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HEAT AND THERMODYNAMICS 



Published by the 

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NewYork 

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HEAT 



AND 



THERMODYNAMICS 



BY 



F. M. HAETMAO 

n 

ment of Physics, and Electrical and 
Cooper Union Day and Night Schools 



t) 
In charge of the Department of Physics, and Electrical and Mechanical Engineering. 



McGRAW-HILL BOOK COMPANY 

239 WEST 39TH STREET, NEW YORK 
6 BOUVERIE STREET, LONDON, E.C. 

1911 



\9 



*>V 



Copyright, 1911 

BY 

McGRAW-HILL BOOK COMPANY 



!*<&> 



©CU305235 

no. r 



PREFACE 



With so many good works in existence, both on Heat and 
Thermodynamics, it may perhaps appear presumptuous to 
publish the following text. The author, however, has long felt 
the need of a text, in teaching the subject of thermodynamics, 
which properly covers, without introducing too much material, 
the fundamental principles of heat measurements. To expect 
an average student to cull from his text book on physics, or 
some treatise on heat, no matter how well the subject may have 
been taught, an introduction to thermodynamics is, in general, 
expecting somewhat more of him than he can accomplish. But 
it has been found, by experience, that a short course on the 
fundamental principles of heat, given as an introduction to the 
subject of thermodynamics, greatly reduces the difficulties, 
experienced by most students, in pursuing this subject. 

Since it is almost impossible for a student to understand a 
complex piece of apparatus, unless he can actually examine it, 
long and tedious descriptions have been purposely avoided. 
Likewise, for the reason that photographs are seldom, if ever, 
of any value, all pictorial illustrations are diagrammatic. 

It is, of course, impossible to teach the subject of thermo- 
dynamics without the application of differential and integral 
calculus; but the aim has been throughout to keep within the 
bounds of elementary mathematics. However, a fair knowledge 
of the calculus, on the part of the reader, has been assumed. 

Very few teachers, if any, can present an unbiassed view of a 
speculative theory; furthermore, before a student has thor- 
oughly mastered the groundwork of any subject, he is not in a 
position to properly discriminate between the various arguments 
that may be advanced, either for or against a speculative theory. 



vi PKEFACE 

It must also be remembered that the average student looks 
upon his instructor as an infallible authority; and that he accepts 
a theory on the mere say so of his instructor, no matter how 
flimsy the arguments upon which it may be based. How fre- 
quently one meets those who are in a condition so deplorable 
that they can talk very glibly about electrons, ionization, etc., 
and are driven helplessly into a corner by one or two well 
directed questions. Whether there is or is not such a thing 
as an atom has nothing to do with the law of definite propor- 
tion. Facts will always remain and theories change to fit them. 
It is for these various reasons that speculative discussions, such 
as that of the kinetic theory of gases, have been avoided, and that 
very hypothetical medium — the ether — has found no place in 
this text. It cannot be too strongly emphasized that before we 
teach metaphysics to a student we must first give him a thor- 
ough training in mathematics and physics. 

It is the author's opinion that the best that can be done in 
any technical course is to thoroughly teach the fundamental 
principles underlying the subject, and that it is impossible to 
give a training which makes the student a practical engineer. 
This part must be learned in practice, and the engineer must 
keep up to date and in proper touch with his profession by reading 
the current engineering literature, and by studying individual 
problems as they arise. 

It must not be understood that the author expects this text 
to supersede such admirable works as Peabody's treatise on 
" The Thermodynamics of the Steam Engine," Zeuner's " Tech- 
nische Thermodynamic " etc.; but rather as a proper prepara- 
tion for the reading of such works. Finally, the author cannot 
express his feelings too strongly in regard to the pleasure he 
experienced, as a student, while reading TyndalPs " Heat a 
Mode of Motion," and Ewing's " The Steam Engine and other 
Heat Engines." 

Thanks are hereby expressed to Mr. Albert Goertz for the 
care with which he read the manuscript. 

F. M. H. 
Cooper Union, July, 1911. 



TABLE OF CONTENTS 



HEAT 

CHAPTER PAGE 

I. Temperature and Thermal Units 1 

II. Calorimetry 11 

III. Production of and Effects of Heat 21 

IV. Expansion of Solids and Liquids 34 

V. Fundamental Equations of Gases 41 

VI. Elasticities and Thermal Capacities of Gases 68 

VII. Propagation of Heat 80 

THERMODYNAMICS 

VIII. Fundamental Principles ....-! 93 

IX. Steam and Steam Engines 116 

X. Entropy 128 

XI. Applications of Temperature-Entropy Diagrams 137 

XII. Elementary Steam and Engine Tests 151 

XIII. Compound Engines 180 

XIV. Internal Combustion Engines and Fuels 192 

XV. Ideal Coefficient of Conversion and Elementary Tests . . . 210 

XVI. Compressed Air and Compressors 230 

XVII. Refrigeration 265 

XVIII. Steam Turbines. 289 

vii 



HEAT 



CHAPTER I 
TEMPERATURE AND THERMAL UNITS 

1. The fundamental conception of hotness or coldness is one 
of bodily sensation. That is, an object is said to be hot or cold 
depending upon whether it gives us one sensation or another 
when we are near it or in contact with it; and the more intense 
the sensation, the hotter or colder the object is said to be. Exper- 
ience, however, teaches us that the estimates so formed are not 
accurate; since the intensity of the sensation experienced, in 
any given case, depends not only upon the condition of the body 
under consideration, but very largely upon our experience imme- 
diately preceding. 

Observation shows that, in general, as bodies are heated or 
cooled they change in volume; and in most cases, other things 
being equal, bodies increase in volume when heated and decrease 
in volume when cooled. Observation further shows that there 
is a continual exchange of heat among bodies; i.e., if in any system 
of bodies some are gaining heat, others are losing heat. Or, in 
other words, there is a continual tendency toward equilibrium. 

2. Temperature. Assume now, that we are dealing with 
two bodies, A and B, and that it is desired to determine which 
is the hotter. To do this, some test specimen, upon which 



2 HEAT 

previous observation has shown a continuous expansion with 
continued application of heat, may be used in the following 
manner: The test specimen is put into contact with A, and after 
a suitable interval of time its length, say, is accurately measured; 
it is then put into contact with B, and its length is again measured. 
If the length is now greater than it was before, i.e., if the test 
specimen expanded when put into contact with B, after having 
been in contact with A, B is hotter than A; for, by previous 
observation it was found that the test specimen expanded con- 
tinually as it became hotter. The body B, therefore, was capable 
of imparting more heat to the test specimen than the body A 
could impart to it, and B is said to be at a higher temperature 
than A . The difference of temperature, then, between two bodies 
may be measured by the amount of change in dimensions which 
a test specimen undergoes when, after having been in contact 
with one of the bodies, it is brought into contact with the other 
body. Such a test specimen is called a thermometer; and, for 
accurate measurements, the nature of the thermometer must be 
such that no appreciable change is brought about in the body 
whose temperature is sought. 

If the two bodies, A and B, of the previous discussion, are now 
brought into contact, and after a suitable interval of time the 
test specimen is put into contact with A and then with B, there 
will be no change in its dimensions; i.e., the two bodies are in 
thermal equilibrium, or, in other words, at the same temperature. 
But, the body B will have lost heat and the body A will have 
gained heat; hence, when a body has the capability of imparting 
heat to another body, it is said to be at a higher temperature. 

Solids, liquids, or gases may be employed in the construction 
of thermometers; but it is important that the substance used does 
not change its state during the change of temperature. For, 
the rate at which a body changes in volume, with respect to 
change in temperature, depends upon its physical state; i.e., 
though a substance may exist in the three different states, its rate 



TEMPERATURE AND THERMAL UNITS 3 

of expansion will, in general, be entirely different in the various 
states; being usually the greatest for gases and the least for solids. 
The rate of expansion, in general, changes abruptly in passing 
from the solid to the liquid, and from the liquid to the gaseous 
state. 

The three substances most generally employed in the con- 
struction of thermometers are: Mercury, alcohol, and dry air; the 
most convenient and most commonly used being mercury. 

There are two standard temperatures, arbitrarily chosen, upon 
which all thermometric scales are based; the one is that of melting 
ice, and the other that of the vapor of boiling water under a pres- 
sure of one standard atmosphere. The pressure of one atmosphere 
being taken equal to that of a column of mercury, at the tempera- 
ture of melting ice, whose height is 76 cm., at 45° latitude and 
sea level, where the acceleration of gravity is 980.60 cm. per sec. 
per sec; or, in c.g.s. units, a pressure of 1.01325 X10 6 dynes per 
square centimeter. 

Thermometric Scales and Thermometers 

3. There are three thermometric scales in use: The centigrade 
scale, the zero of which is the melting-point of ice, and the point 
corresponding to the temperature of the vapor from boiling 
water under standard conditions, called the boiling-point, is 
marked 100. Hence there are 100 units, called degrees, for the 
interval between the melting-point and the boiling-point. The 
Fahrenheit scale is marked 32 for the melting-point and 212 for 
the boiling-point; hence, there is an interval of 180 degrees 
between the two fixed points. The Reaumur scale is marked 
for the melting-point and 80 for the boiling-point; hence, there is 
an interval of 80 degrees between the two fixed points. Fig. 1 
is a diagrammatic representation of the relation of the three 
scales. From the foregoing it is obvious that the value of a 
degree on the Fahrenheit scale is 5/9 of a degree on the centi- 



HEAT 



100 i 


212 


Ujl 


h 


Ql 


Ul 


<l 


I 


01 © 

2 § 
•- 


z 

LU 


z> 


I 


U| 


< 


0| 


u. 


o ! 


32 



80 



grade scale, and the value of a degree on the Reaumur scale is 
5/4 of a degree on the centigrade scale. Since both the centi- 
grade and Fahrenheit scales are in 
common use, it is convenient to 
have a simple method of conversion 
from one scale to the other. 

Let it be desired to convert the 
temperature on the centigrade 
scale to the Fahrenheit scale. Since 
one degree centigrade equals 9/5 
degree Fahrenheit, it follows that 
an interval of 6°C. is equal to an 



8 



Fig. 1. 



interval of — 6°F. ; but, since the 



melting-point on the Fahrenheit scale is marked 32, we must add 
32 to give the Fahrenheit reading. Therefore, to convert a 
temperature on the centigrade scale to the Fahrenheit scale, we 
must multiply the reading by 9/5 and add 32. And similarly, to 
convert Fahrenheit to centigrade we must subtract 32 from the 
reading and multiply by 5/9. 

A method of conversion, which is simpler, is to solve for 
that temperature for which both scales read the same. This 
temperature is obviously below zero, and is therefore negative. 

9 
Let be that temperature on the centigrade scale; then, —0+32 

5 



is the reading on the Fahrenheit scale, 
are by condition equal. Hence, 



But these two readings 



from which 



= |0+32; 



0=-4O. 



Therefore, to convert from one scale to the other, the most con- 
venient way is to add 40 to the reading, multiply this by the con- 



TEMPERATURE AND THERMAL UNITS 5 

version factor and deduct 40. To convert from centigrade to 
Fahrenheit, add 40 to the reading, multiply by 9/5 and deduct 
40. To convert from Fahrenheit to centigrade, add 40, multiply 
by 5/9 and deduct 40. 

4. Mercurial Thermometer. The mercurial thermometer con- 
sists of a capillary tube of, as nearly as obtainable, uniform bore, 
on one end of which is blown a bulb. The bulb is filled with 
mercury and heated so as to drive out all the air. When this 
has been satisfactorily performed, so that nothing but mercury 
remains in the bulb and tube, the tube is hermetically sealed. 
The bulb and tube are then immersed in ice from which the water 
is allowed to drain away, and the point to which the mercury 
falls marked on the stem. Next, the thermometer is immersed 
in saturated steam, under standard pressure, and the point to 
which the mercury rises marked on the stem. It is, however, 
necessary to allow considerable time to elapse between the seal- 
ing of the tube and the determination of the fixed points; since 
glass, after having been heated, does not, upon being cooled, 
immediately return to its original volume. If the bulb contracts 
after the fixed points have been placed on the stem, the ther- 
mometer will read too high. Joule found that the bulb of a certain 
thermometer, upon which he had taken observations for twenty 
years, was still changing slightly at the end of that time. 

After the fixed points are determined, a thread of mercury is 
detached, and, by means of it, the tube calibrated to the desired 
scale. In this way the units on the scale represent equal volumes, 
and not necessarily equal lengths. But in good thermometers 
the tube is of so nearly uniform bore that the lengths of a degree 
do not differ by any considerable amount over different parts 
of the scale. 

Since glass changes in volume when its temperature changes, 
it is obvious that the indications of the mercurial thermometer 
are proportional to the relative changes between the mercury 
and glass, and do not necessarily indicate the true changes in 



6 HEAT 

temperature; i.e., the indicated temperatures between the two 
fixed points depend upon the substances used in construction. 
It is readily seen that two mercurial thermometers, if constructed 
of different qualities of glass, which have not the same rates of 
expansion with respect to mercury throughout the entire scale, 
will differ slightly in their readings for some parts of the scale, 
even though they read alike for the fixed points. 

5. Alcohol Thermometer. The alcohol thermometer is con- 
structed in a manner similar to the mercurial thermometer. Its 
chief advantage lies in the fact that it may be used for temperatures 
below the melting-point of mercury, which is — 39°C. However, 
on account of its low boiling-point, which is 78.2°C, alcohol 
cannot be employed for high temperatures; on the other hand, 
the boiling-point of mercury is 357°C. 

The discussion of the air thermometer will be deferred until 
after the discussion of the laws of gases. 

6. Thermo Couple. When the junction of two dissimilar 
metals is heated, an e.m.f. is developed; and since this e.m.f. 
is a function of the temperature, such a combination, called a 
thermoelectric couple, or simply thermo couple, may be employed 
to indicate temperatures. The thermo couple is, in many cases, 
where a bulb thermometer cannot be employed, a very conven- 
ient device for measuring changes of temperature; and it is par- 
ticularly valuable in enabling us to estimate changes of temper- 
ature above the boiling-point of mercury. By employing proper 
metals, such as platinum and iridium, very large ranges of tem- 
perature can be measured; the melting-point of iridium being 
about 2500°C. and that of platinum about 1775°C. 

7. Resistance Thermometer. The fact that the ohmic resist- 
ance of a metal is a function of its temperature makes it possible 
to estimate changes in temperatures, by noting the changes in 
resistance of a particular conductor employed for the purpose. 

From the foregoing, it is obvious that if any body or combina- 
tion of bodies manifest some change, which is a function of the 



TEMPERATURE AND THERMAL UNITS 7 

temperature and readily measurable, such body or combination 
of bodies may be employed to indicate temperatures. But it 
is to be carefully noted that the device employed must be such 
that the temperature of the body, upon which the measurements 
are made, is not appreciably altered by the test body; and that, 
in any case, the changes produced in the test body are peculiar 
to it, and not necessarily proportional to the changes that would 
be produced in some other instrument of a different type. There- 
fore, temperatures must always be referred to some scale chosen 
as a standard. This will be dealt with more fully in the discussion 
of thermodynamics. 

Heat as a Measurable Quantity 

8. If a quantity of water at a temperature ti, be mixed with 
an equal quantity of water at a temperature T2, the resulting 
temperature of the mixture is very nearly the mean between the 
two initial temperatures. If it requires a certain quantity of 
heat, q, to raise n grams of water through a given temperature 
interval, then it obviously requires a quantity of heat, mq, to raise 
mn grams of water through the same temperature interval. If 
the water be cooled through the same temperature interval, then 
there is imparted to the surrounding bodies a quantity of heat 
which is equal to that absorbed by the water while being raised 
through that temperature interval. 

If the quantities of heat, required to raise a given mass of 
water through equal temperature intervals throughout the chosen 
thermometric scale, were all equal, then the resulting temperature 
obtained when mixing equal masses of water would be the exact 
mean between the two initial temperatures. This is shown by 
experiment to be very nearly, but not quite true. Hence, there 
is not strict proportionality, in the case of water, between change 
of temperature, according to our thermometric scale, and change 
of heat. 



8 HEAT 

If equal masses of water and some other substance, say, mercury, 
be mixed, the resulting temperature will differ considerably from 
the mean between the two initial temperatures. Experiment 
shows that if mercury at a temperature ti, be mixed with water 
at a temperature T2, the mass of the mercury must be 29.85 times 
the mass of the water so as to give a resulting temperature equal 

T1+T2 

to -3-. 

9. Thermal Capacity. The thermal capacity of a body is 
numerically equal to the ratio of change in heat to the corresponding 
change in temperature produced by it. The preceding paragraph 
states that water has, mass for mass, a greater thermal capacity 
than mercury in the ratio of 29.85 : 1; or the thermal capacities 
of equal masses of mercury and water are to each other as 0.0335 : 1. 
If copper be compared with water the ratio is found to be as 
0.0933 : 1. In general, the ratio is less than unity, has different 
values for different substances, and varies somewhat with change 
of temperature. One notable exception is hydrogen gas, where 
the ratio is found to be, at constant pressure, as 3.409 : 1; and 
at constant volume, as 2.42 : 1. 

To compare different quantities of heat, it is necessary to 
choose some substance as a standard. On account of convenience, 
water has been so chosen. 

10. The Calorie. The quantity of heat required to raise the 
temperature of 1 kilogram of water through 1 degree centi- 
grade, is called a calorie, and is the unit adopted for heat measure- 
ments. But, since this quantity varies slightly for different 
temperatures it becomes necessary, in making exact measure- 
ments, to specify some particular quantity. There are three 
different definitions for the calorie: 

(1) The quantity of heat required to raise the temperature 
of 1 kilogram of water from 0°C. to 1°C, called the zero 
calorie. 

(2) One hundredth part of the heat required to raise the tern- 



TEMPERATURE AND THERMAL UNITS 9 

perature of 1 kilogram of water from 0°C. to 100°C, called 
the mean calorie. 

(3) The quantity of heat required to raise 1 kilogram of 
water from 15°C. to 16°C, called the common calorie. 

Since it is impossible to realize accurately the first or second 
of these, on account of the difficulty experienced in working with 
water at 0°C, the last, or common calorie, is the one most gen- 
erally used. Furthermore, due to the fact that a great many 
heat measurements are made in the range between 15°C. and 25° 
C, no large corrections for change in thermal capacity, due to 
change in temperature, when the common calorie is employed, 
are necessitated; hence, this unit is more convenient than the 
others. 

Since, in ordinary heat measurements, masses are generally 
specified in grams, a secondary unit, called gram calorie, having 
the gram instead of the kilogram for the unit mass, is usually 
found more convenient than the calorie. In what follows, unless 
otherwise specified, by calorie is to be understood the quantity 
of heat required to raise the temperature of 1 kilogram of water 
from 15°C. to 16°C, and by gram calorie, one thousandth part 
of the calorie. 

11. British Thermal Unit. The thermal unit most commonly 
employed in engineering practice, in England and America, is 
the British Thermal Unit or B.T.U.; it is the quantity of heat 
required, at ordinary temperatures, to raise the temperature of 
1 pound of water through 1 degree Fahrenheit. 

12. Thermal Capacity per Unit Mass and Specific Heat. The 
ratio of the quantity of heat required to raise the temperature 
of a given mass of a substance through a given temperature 
iriterval, to the quantity of heat required to raise an equal mass of 
water through the same temperature interval, is called the specific 
heat of the substance. And, since the unit of heat — the gram 
calorie — is the quantity of heat required to raise the temperature 
of 1 gram of water through 1 degree centigrade, it follows 



10 HEAT 

that the quantity of heat, measured in gram calories, required 
to raise the temperature of 1 gram of a substance 1 degree 
centigrade, is numerically equal to the specific heat of the sub- 
stance; or, in other words, the specific heat of a substance is 
numerically equal to its thermal capacity per unit mass. 

13. Water Equivalent. By the water equivalent of a body 
is understood the mass of water which has a thermal capacity 
equal to that of the given body, and is numerically equal to the 
mass of the body multiplied by its thermal capacity per unit 
mass. 

14. Method of Mixtures. Assume a mass of water mi, at 
a temperature ti, to be mixed with a mass wi2, of some other sub- 
stance, at a higher temperature T2, yielding for the mixture a 
resulting temperature of 6. Then, if no heat is lost to or gained 
from the surroundings, during the operation, and the thermal 
capacities of the water and substance are sensibly constant for 
the temperature ranges experienced, it follows that, since the heat 
gained by the water is equal to that lost by the substance, we 
must have 

mi(0— Ti)=ra 2 c(T 2 — 0); (1) 

where c is the thermal capacity per unit mass of the substance, 
amd m2C is its water equivalent. 

To take a numerical example, assume 268 grams of water at 
a temperature 10°C. to be mixed with 1000 grams of mercury at 
a temperature 100°C, giving a temperature of 20°C. for the mix- 
ture. Substituting in equation (1), we have 

268(20 - 10) = 1000c(100 - 20) ; 

from which, the thermal capacity of mercury, per unit mass, is • 

268(20-10 ) . 
C ~ 1000(100-20) - UM66b - 



CHAPTER II 
CALORIMETRY 

15. As thermometry has for its object measurement of tem- 
peratures, so has calorimetry for its object the measurement of 
quantities of heat. 

In equation (1), Art. 14, it was shown what must be the 
relation between the masses involved, the changes in temperature, 
and the thermal capacity per unit mass of a substance, when 
two substances are mixed and assume a common temperature. 

The actual determination of thermal capacities is, however, 
not so simple. Since, in general, the vessel in which the mixing 
takes place suffers a change in temperature, its thermal capacity 
must be taken into consideration. Furthermore, there is usually 
an exchange of heat between the vessel, in which the experiment 
is performed, and the surrounding medium during the progress 
of the experiment. The vessel, specially designed, in which the 
mixing takes place, is called a calorimeter. It therefore follows, 
from what has just been said, that in making heat measurements 
it is necessary to know not only the thermal capacity of the calo- 
rimeter, but also, the rate at which, for a given difference of tem- 
perature, exchange of heat takes place between the calorimeter 
and the surrounding medium. 

16. Law of Cooling. The rate at which a body loses heat to 
surrounding bodies is independent of its thermal capacity, and 
depends upon the nature and area of its exposed surface, and the 
difference in temperature between it and the surroundings. For 
small differences of temperature, i.e., up to a difference of about 

11 



12 HEAT 

15°C, from ordinary room temperature, the rate of cooling is 
very nearly proportional to the difference of temperature. This 
is known as Newton's Law of Cooling. The foregoing then states 
that the rate at which a body loses heat at any instant is a func- 
tion of its surface, and of the difference of temperature between 
it and surrounding bodies. Newton's law may be stated as 
follows: 

§--'*■> 0) 

where Q is quantity of heat, t time, K some constant, depending 
upon the surface of the body, and t the difference of temperature. 
If m is the mass of the body, and c the thermal capacity per unit 
mass, then equation (1) may be written 

4r- & - • • w 

If c is constant, then equation (2) becomes 







dt ~™ ; 


where k= — . 
mc 


Separati 


ing the variables, we find 
— = — kdt, 


from which 




log f r -kt; 


and 




T=fcis" te . 



(3) 

To determine the constant of integration A>i, assume that we 
begin to reckon time when t=ti; i.e., t=ti when £ = 0. Making 
this substitution, in equation (3), we find &i=ti; hence, finally 

T=Tl£-^ (4) 



CALORIMETRY 13 

If m, c, and K are known, thus fixing the value of k, then, by 
means of equation (4), the difference of temperature, t, between 
the body under consideration and the surroundings at any time, t, 
can be predicted, provided the initial difference of temperature 
be known, and the surroundings remain constant in temperature. 

It is found, by experiment, that bodies having highly polished 
surfaces, other things being equal, lose heat less rapidly than do 
those having rough dark surfaces; hence, calorimeters should 
have their exposed surfaces highly polished. Furthermore, the 
thermal capacities of calorimeters should be small in comparison 
with those of the bodies contained in them, and upon which 
measurements are being made. Also, while the experiment is 
under progress, the calorimeter should be protected from draughts 
of air. We are not as yet in a position, nor is it essential, to enu- 
merate all precautions that must be taken to give results of abso- 
lute precision. 

17. Thermal Capacity of a Calorimeter. Since the calo- 
rimeter in which the bodies, upon which measurements are to be 
made, are contained, always suffers a change in temperature, 
it is necessary to know its thermal capacity. But, since the thermal 
capacity of the calorimeter, in general, is small in comparison with 
that of the bodies upon which measurements are being made, 
it follows that a small error in the determination of the thermal 
capacity of the calorimeter will not seriously affect the results 
obtained for these bodies. 

The thermal capacity of a calorimeter may be obtained, 
though not with absolute precision, in the following manner: 
The calorimeter containing water has its temperature noted, 
the water being first stirred to insure uniform temperature, then 
immediately a quantity of water at some other temperature is 
poured into the calorimeter, the contents stirred and the resulting 
temperature is noted. If C is the thermal capacity of the calo- 
rimeter and stirrer, wii the mass of water originally in it, ti the 
temperature of calorimeter and contents before mixing, and 6 



14 HEAT 

the common temperature after mixing, then on the assumption 
that ti is higher than 0, the loss in heat, suffered by the calorimeter 
and water originally in it, is 

(mi+C)(T!-0). 

The gain in heat, by the water poured into the calorimeter, is 

m 2 (e-T 2 ); 

where ra 2 is the mass, and t 2 the temperature of the water poured 
into the calorimeter. But, if there are no other heat exchanges, 
the loss on the one side must be equal to the gain on the other, 
hence 



from which 



(mi+C)(Ti-6)=ra 2 (0-T 2 ); 



c= m 2 (0^T 2 )_ 

Tl-6 • 



There will always be an exchange of heat between the calo- 
rimeter and surroundings; but this can be reduced to a small 
quantity by choosing the masses of water such that the result- 
ing temperature of the calorimeter is as much below the room 
temperature as was its initial temperature above the room tem- 
perature. When it is possible, large differences of tempera- 
ture, between the calorimeter and room, should be avoided. 

18. Cooling Constant of a Calorimeter. When it is impossible 
to have the initial and final temperatures differ by equal amounts 
from the room temperature — one, of course, being above and the 
other below — then, to obtain accurate results, correction must 
be made, as the case may be, for loss or gain in heat. To do this, 
the calorimeter is filled with water, at about 15°C. or 16°C. 
above room temperature, to the same height as it will be when 
the experiment proper is performed. The temperature is then 



CALORIMETRY 15 

noted at short intervals of time, the water being continuously 
stirred to insure a uniform temperature throughout at any instant. 
If the room temperature has remained constant during the 
progress of the experiment, then, obviously, equation (4), Art. 16, 
applies and the constant k of this equation is determined. In 
general, however, better results are obtained by plotting the 
observations; using times as abscissas and differences in temper- 
atures, between the calorimeter and the room, as ordinates, and 
passing a smooth curve through the points so found. The slope 
of the tangent then, to the curve at any point, is the rate of change 
of temperature at that point; and this slope, divided by the dif- 
ference of temperature, or in other words, by the ordinate of the 
point, is, according to Newton's law of cooling, a constant for any 
point on the curve. Drawing a number of tangents and divid- 
ing the slope of each by its ordinate, will give quotients nearly 
equal; and the mean of these quotients will be a fair value for 
the rate of change of temperature for unit difference of temper- 
ature. The rate so obtained multiplied by the thermal capacity 
of the calorimeter and contents is numerically equal to the quan- 
tity of heat lost, by the calorimeter, per unit time per unit differ- 
ence of temperature. 

19. Determination of Thermal Capacities. Thermal capacities 
may be determined in various ways; the simplest, though not 
necessarily the most accurate, and not applicable in all cases, is the 
method of mixtures. The substance, whose thermal capacity is 
sought, is heated to a temperature ti, which is noted. It is then 
immediately transferred to a calorimeter of thermal capacity C, 
containing a mass of water m 2 , at a temperature T2. If the result- 
ing temperature is 6, and no heat has been lost to or gained from 
the surroundings during the operation, then the heat lost by the 
one side must be equal to the heat gained by the other; hence we 
have 

mic(Ti-0) = (m 2 +C)(0-T 2 ); . . . . . (6) 



16 HEAT 

where mi is the mass of the substance, and c its thermal capacity 
per unit mass. From equation (6), we find 

c = (m 2 +C)(e-T 2 ) ...... (7) 

wi(ti-g) ' v ' 

which determines the thermal capacity per unit mass of the 
substance. If the resulting temperature differs materially from 
the room temperature, then corrections will have to be made, from 
the curve of cooling of the calorimeter. 

20. Method of Cooling. If heat is generated or absorbed 
when two substances are mixed, the thermal capacity of a sub- 
stance cannot be determined by the method of mixtures. As 
an example, if sulphuric acid is mixed with water, considerable 
heat is evolved; hence, recourse must be had to some method, 
other than the method of mixtures, in determining the thermal 
capacity of sulphuric acid. This may conveniently be done by 
what is known as the method of cooling. 

Assume that we have a calorimeter of known thermal capacity, 
C. Let the calorimeter be filled to a definite height with water, 
and the time noted which is required for the calorimeter and 
contents to cool, from a temperature ti to a temperature t 2 , 
when exposed to a definite and constant room temperature. 
Next, the calorimeter is filled to the same height with the liquid, 
whose thermal capacity is sought, and the time which is required 
to cool from ti to t 2 , when the calorimeter and contents are sub- 
jected to precisely the same conditions as when filled with water, 
is again noted. Then, since the average difference of temperature 
between calorimeter and contents and the surroundings is the 
same in both cases, it follows that the thermal capacities, in the 
two cases, are to each other directly as the times required in cooling 
through the same temperature intervals. Therefore, if h is the 
time required for the calorimeter and water to cool from ti to 
T2, and t2 is the time required for the calorimeter and the sub- 



CALORIMETRY 17 

stance whose thermal capacity is sought, to cool through the same 
temperature interval, it follows that 

C+mi C+m 2 c , Q . 

-ir = -~h"' (8) 

where wi is the mass of water, W2 the mass of the liquid, and c 
its thermal capacity per unit mass. From equation (8), since all 
quantities excepting c are known, the thermal capacity per unit 
mass is determined. 

It is, however, not necessary to know the thermal capacity of 
the calorimeter. For, if the calorimeter be first cooled through 
the temperature interval, ti— T2, when empty, then when filled 
with water, and again when filled with the liquid whose thermal 
capacity is sought, and the times t, h, and fe are noted, we shall 
have 

C C+mi C+7112C 



t h t 2 



(9) 



where t is the time required when empty, t\ when filled with water, 
and h when filled with the liquid whose thermal capacity is sought, 
and the other symbols having the same significance as before. 
Eliminating C, from equation (9), we find 

c=^=| (10) 

A convenient form of apparatus, for the method of cooling, 
is an alcohol thermometer with its bulb greatly enlarged and 
having the form of a hollow cylinder. The substance is then 
placed directly inside of the bulb and the whole thermometer, 
while cooling, exposed to a constant temperature; this may 
readily be brought about by placing the thermometer inside of a 
vessel which is surrounded by melting ice. 

21. Mechanical Equivalent of Heat. From mechanics we 
have the following statement: " The change in kinetic energy 



18 HEAT 

that a body undergoes in passing over a given path is equal to 
the work done in traversing that path." The foregoing statement 
is very simple, and readily comprehended when considered in the 
purely mechanical sense. That is, it is merely stated that when- 
ever a given amount of kinetic energy, has been destroyed or 
developed in a system, an equivalent amount of work has been 
done by the system or, as the case may be, on the system. The 
statement, however, does not concern itself with the transforma- 
tion of one form of energy into another. 

The most general experience, common to all, is that of the 
destruction of energy, in the form of mechanical motion, by fric- 
tion or impact and the simultaneous evolution of heat. However, 
it was not until 1842 that a clearly formulated statement was 
made, by J. R. Mayer, to the effect, that when heat is converted into 
work, or vice versa, the ratio of the numbers representing the two 
quantities involved is constant. Unfortunately, the figures upon 
which Mayer based his calculations were in error, and consequently 
the value obtained for the mechanical equivalent was also in 
error. 

Shortly subsequent to Mayer's enunciation, Joule began his 
series of experiments to determine the mechanical equivalent 
of heat by direct measurement. Joule's method was essentially 
as follows: A vessel having fixed vanes, was filled with water, 
and a paddle-wheel was caused, by means of falling weights, to 
rotate in the water. The fixed vanes prevented the water from 
assuming a rotary motion. The heat developed manifested 
itself by a rise in temperature, and the work done was measured 
by the weights and distance fallen. A series of experiments was 
then made, using mercury instead of water. Another series of 
experiments was made by causing one iron plate to rotate with 
friction over another iron plate under water. 

It is, of course, understood that experiments like these are 
attended by great difficulties, and various precautions must be 
taken and corrections made which cannot here be enumerated. 



CALORIMETRY 19 

Still, Joule obtained fairly consistent results; and, the figures he 
finally published were not greatly in error. 

The results were expressed in meter-kilograms of work per 
calorie; i.e., Joule found when using water that it required 423.9 
m.kgs. of work to develop one calorie, 424.7 m.kgs. of work per 
calorie when using mercury, and for the experiment with iron, 
the number was found to be 425.2. It must be remembered 
that, if comparisons are to be made between results obtained by 
experiments, which have been performed in different localities, 
corrections will have to be made for variations in the value of g. 

Rowland varied Joule's method by using a motor drive instead 
of falling weights. The vessel was suspended and the torque 
required to prevent rotation measured. This enabled a much 
more rapid expenditure of energy, and a consequent rapid rise 
in temperature; thus making the correction due to cooling much 
smaller. Rowland's experiment covered the range from 5°C. 
to 36 °C. Since the thermal capacity of water varies for different 
temperatures, a variation was found for the mechanical equivalent 
of heat. Rowland found, at Baltimore, where g = 980.0, for the 
mechanical equivalent of heat, of the common calorie, 427.3 
m.kgs. This may be taken as being substantially correct. 

Anthony modified Rowland's method by having a continuous 
flow of water through the calorimeter, the temperature of the 
inflowing water was constant, and its rate of flow was so regulated 
that the vessel was always at room temperature. The mass 
of water flowing for a given time was determined by weighing, 
the number of rotations made by the paddle was recorded on a 
speed counter, and the torque was noted, together with the tem- 
perature of the inflowing and outflowing water. By this method, 
gince the vessel is always at room temperature, no corrections 
for cooling are required; and furthermore, since the vessel suffers 
no change in temperature, its thermal capacity need not be known. 
The values obtained by this method were in practical concordance 
with those found by Rowland. 



20 HEAT 

In the English system of units, the value for the mechanical 
equivalent of heat, usually employed, is 778 ft. -lbs. per B.T.U. 

Various experiments have been performed, by sending an elec- 
tric current through a conductor wound upon an insulating support 
and submerged in water. By noting the current, the applied 
e.m.f., and the time, during which the current has been flowing, 
the energy input is readily computed. Results obtained by this 
method agree almost precisely with those obtained by the methods 
previously described. 

A safe value to use, which if in error is only slightly so, and 
which is correct for all practical purposes, is 4.195 X10 7 ergs per 
common gram calorie. 

It will now readily be seen that, since we can measure electrical 
quantities with great precision, the best method for obtaining a 
definite quantity of heat is by sending a steady current for a 
given time through a given resistance; it being remembered that 
there are alloys whose resistances are practically independent 
of temperature. 



CHAPTER III 
PRODUCTION OF AND EFFECTS OF HEAT 

22. Since, in a great many cases, those changes which evolve 
heat during their progress, require the application of heat to bring 
about a change in the reverse order, it is inadvisable to consider 
the production of heat and the effects of heat independently. 

Whenever a change of a chemical nature takes place there is 
either an absorption or a liberation of heat. In general, heat is 
absorbed when a compound is split up into the elements com- 
posing it; and heat is liberated when the elements recombine 
to form the compound. Those compounds which evolve heat, 
during their formation, are called exothermic compounds; and 
those rare compounds which absorb heat, during their formation, 
are called endothermic compounds. 

It will now be instructive to consider some particular substance 
and the various changes which take place with the continuous 
application of heat. Suppose that we are dealing with a definite 
mass of ice, under a given pressure, whose temperature is below 
that of its melting-point for the applied pressure. (The melting- 
point of ice changes slightly with change of pressure; i.e., the 
melting-point is lowered about 0.0075°C. for each increment in 
pressure equal to one atmosphere.) A definite amount of heat 
then, must be applied to raise the temperature of the ice to the 
melting-point. If heat be then further applied the temperature 
will no longer change; but, a change of physical state takes place 
together with a continuous absorption of heat until all the ice is 
melted. 

21 



22 HEAT 

23. Heat of Fusion. The quantity of heat required to convert 
unit mass of a solid into the liquid state without change of tem- 
perature is called the heat of fusion of the substance. In the case 
of ice, the heat of fusion is approximately 80 gram calories per 
gram. If now, after all the ice has been converted into water, 
heat be continuously applied, the temperature will rise progress- 
ively until, if the liquid be under a pressure equal to one stand- 
ard atmosphere, the temperature of 100 °C. is reached. The 
temperature will then cease to rise, provided the pressure be 
maintained constant, and a change of physical state, namely, 
vaporization at constant temperature, takes place progressively 
with the continuous application of heat until all the water is 
evaporated. 

The heat of fusion of ice may be determined as follows: Let 
L, in, and ti, respectively, represent the. heat of fusion, the mass, 
and the initial temperature of the ice; and M, and 12, respectively, 
represent the water equivalent of calorimeter and contents, and 
the initial temperature of calorimeter. Let the ice now be sub- 
merged in the water in the calorimeter until it is all melted, and 
the calorimeter and total contents assume a common temper- 
ature 6. The total heat, then, consumed by the ice in having its 
temperature raised from ti to 0, being converted into water at 
this temperature, and in raising the temperature of the liquid 
from to 6, is 

racTi+ra(L+0); 

where c is the thermal capacity per unit mass of ice. But this 
quantity of heat must be equal to 



hence, 
From which 



M(t 2 -0); 

racTi+ra(L+6) =M(t 2 -8), 



L = -(T 2 -6)-(cTi+e). 
m 



PRODUCTION OF AND EFFECTS OF HEAT 23 

24. Heat of Vaporization. The quantity of heat required 
to convert unit mass of a liquid into a vapor without change of 
temperature is called the heat of vaporization of the substance. 
In the case of water, at 100°C, the heat of vaporization is approx- 
imately 537 gram calories per gram. The heat of vaporization 
for most substances becomes less as the temperature rises. 

The heat of vaporization of water may be determined by the 
method of mixtures as follows : Let superheated steam, at a tem- 
perature ti, be passed into a calorimeter containing water at a 
temperature T2. This is continued until a convenient rise of 
temperature is obtained in the calorimeter, and a mass of steam 
m has been condensed. The quantity of heat, given up by the 
steam, is , 

rac(Ti— T)+mr-+m(T — 0); 

where t is the temperature at which the condensation takes place, 
r the heat of vaporization, c the thermal capacity per unit mass 
for superheated steam, and.0 the resulting temperature. But 
this quantity of heat must be equal to 

M(6-T 2 ); 

where M is the thermal capacity of the calorimeter and water 
initially contained in it. Hence, 

rar+ra{c(Ti-T) + (T-6)S=Af(e-T2); 
from which 

r=^(8-T 2 )-{c(T 1 -T)+(T-6)|. 

fit 

To obtain accurate results by means of calorimetric methods, 
as previously stated, it is always necessary to take certain pre- 
cautions and apply proper corrections. 

25. Sublimation. Under certain conditions a substance may 
pass directly from the solid to the gaseous state without passing 



24 



HEAT 



through the liquid state. Such a change is called sublimation. 
Substances such as camphor and iodine, when gently heated, 
pass readily from the solid to the gaseous state without melting. 
Ice, under normal pressure, also sublimes at temperatures lower 
than the melting-point. 

26. Superheating. If now, after all of the liquid has been 
converted into vapor, heat be further applied, the temperature 
will rise progressively with continued application of heat, and the 
vapor will become superheated. 

All of the foregoing may, instructively, be represented diagram- 
matically; bearing in mind that the thermal capacities per unit 
mass of water for the three states are: Solid 0.504, liquid 1, and 
gaseous 0.481. 




Fig. 2. 



Let, as in Fig. 2, the temperatures be taken as ordinates and 
quantities of heat as abscissas. Then, if we assume some arbitrary 
zero, such as 0, for the initial condition of the ice, we have for the 
application of the quantity of heat Oe, the increment of tempera- 
ture ea, to the point of fusion. The application of the quantity 
of heat ef then brings about the conversion from the solid to the 
liquid state at constant temperature. The further application 
of the quantity of heat fg brings about the elevation of temper- 
ature, from the temperature of fusion, to that of vaporization. 
The application of the quantity of heat gh brings about complete 



PKODUCTION OF AND EFFECTS OF HEAT 25 

vaporziation at constant temperature. The further application 
of heat brings about superheating, as shown by the line di. 

In general, the physical history of substances with continued 
application of heat will be similar to the process just discussed; 
but, the ratios of the quantities involved will be entirely different 
for each substance. 

Assume now, that the process takes place in the reverse order, 
step by step; i.e., the steam is first cooled to the point of conden- 
sation, then condensation takes place at constant temperature, 
and so on, step by step, until the initial condition is reached. 
Then during each change, heat is liberated precisely equal in 
amount to that which was absorbed when the change was taking 
place in the opposite direction. 

27. Reversible Processes. The changes just described and 
depicted in Fig. 2 are, however, not the only changes involved. 
Assume the applied pressure to be maintained constant throughout 
the entire change, the volume then of the substance will be chang- 
ing continually; and, in general, will be increasing with the tem- 
perature. When the volume is increasing, work is being done 
by the substance in overcoming the applied pressure. When the 
volume is decreasing, work is being done on the substance by the 
applied pressure. From the initial condition up to 4°C, this 
being the temperature corresponding to the maximum density 
of water, work is being done on the substance. For all tem- 
peratures higher than this, the substance expands continuously 
with the continued application of heat; and work is being done 
by the substance in overcoming the applied pressure. When the 
process takes place in the reverse order, then, wherever heat was 
absorbed and work done by the substance during the direct process, 
heat will be liberated by, and work will be done on the substance 
during the reverse process. And if these quantities be mutually 
equal, and no permanent changes have been made to take place 
in the surrounding bodies, by this cycle of operations, either 
process is said to be a reversible process. That is, when a system 



26 HEAT 

undergoes a change, or a series of changes, the process is said to be 
reversible if, after it has taken place, a second process can be made 
to take place, in a manner, such that when the system is again in 
its initial condition, there remain, due to these various changes, 
no changes outside of the system. 

A little consideration will show that vaporization at constant 
pressure, and hence at constant temperature, provided we could 
have perfect insulation and no. friction, would be a reversible 
process. For, under the assumed conditions, the amount of work 
done by the vapor, during its formation, in overcoming the external 
pressure, is precisely equal to the amount of work done on the 
vapor during its condensation. Furthermore, the quantity of 
heat absorbed, during vaporization, from a reservoir of heat at 
constant temperature, is precisely equal in amount to the quantity 
of heat rejected, to the reservoir, during condensation. Hence, 
since all quantities involved balance each other, and no changes 
have been brought about in the surroundings, the process is 
reversible. 

The cooling and heating of a substance at constant pressure, 
together with its consequent changes in volume, can be made 
a reversible process only by the aid of a perfect regenerator. The 
following discussion will make this clear. Assume that a body 
at a temperature t», which is also the temperature of the first 
reservoir, cools to a temperature ti, by being put, successively, 
into contact with n reservoirs, perfectly insulated from each 
other, and each reservoir differing in temperature from the one 
adjacent to it by an amount equal to (x n — Ti)/(n— 1). The* 
body then, in cooling, gives up to each reservoir, excepting 
the first, a definite quantity of heat, and has done upon it, by the 
constant external pressure, a definite amount of work. Let the 
process now take place in the reverse order, i.e., the body at a 
temperature ti is put into contact with the reservoir at a temper- 
ature T2; a definite quantity of heat will be absorbed, which will 
be precisely equal to that rejected when put into contact with 



PRODUCTION OF AND EFFECTS OF HEAT 27 

the reservoir at a temperature ti, after having been in contact 
with the reservoir at a temperature 12. Let this be continued 
until the temperature T n is again reached. Now, the work done 
by the external pressure on the body, while cooling from the tem- 
perature Tn to the temperature ti, is precisely equal in amount 
to the work done by the body, in overcoming the external pressure, 
while being heated from the temperature ti to the temperature 
Tn. This process, however, is not perfectly reversible; since the 
reservoir at the temperature i n has given up heat and received 
none, and the reservoir at the temperature ti has received heat 
and given up none. In the limit, however, as the fraction 
(: n - Ti)/(n— 1), approaches zero for its value, the process becomes 
perfectly reversible. But this implies a perfect regenerator; 
i.e., a series of reservoirs which are perfectly insulated from each 
other, and still have a continuous variation in temperature 
throughout the series; but this is practically impossible. Hence, 
it is obvious that, since in the case of vaporization, we must assume 
no radiation and no friction to make the process reversible, and 
in the case of cooling and heating of a body, we must assume a 
perfect regenerator and no friction and radiation to make the 
process reversible, the process as described and represented 
diagrammatically in Fig. 2 is reversible only in an ideal sense; 
i.e., an ideally reversible process. 

28. Irreversible Processes. If a constant e.m.f. be applied 
to the terminals of a homogeneous conductor, a current will flow 
which is directly proportional to the applied e.m.f., and inversely 
to the resistance of the conductor. After a time the conductor 
will reach a constant temperature; i.e., the rate at which the 
energy is being converted into heat by the conductor, due to its 
resistance, will be equal to the rate at which heat, expressed in 
the same units, is given to the surroundings by the conductor. 
This, however, does not mean that there is thermal equilibrium. 
In this case thermal equilibrium can only be brought about by dis- 
connecting the applied e.m.f., and consequently, discontinuing the 



28 HEAT 

dissipation of energy. This being done, the conductor will finally 
assume the temperature of the surroundings, and thermal equilib- 
rium will have been established. This process, viz, the con- 
version of energy, in the form of an electric current, into energy, 
in the form of heat, differs essentially in one particular feature 
from the process discussed in Art. 27. The former process, 
which we termed an ideally reversible process, can be made to take 
place, barring various losses, in the reverse order. The latter 
process, however, cannot be made to take place in the reverse 
order; i.e., it is absolutely impossible to cause a current to flow in 
a homogeneous conductor by applying heat to it. Such a process 
is called an irreversible process. 

Another example of an irreversible process is that of the con- 
version of energy, in the form of mechanical motion, into heat by 
friction. For it is impossible to restore a system of bodies to 
their initial positions by the application of a quantity of heat to 
the surfaces, equal to that which was evolved, due to friction, 
during their displacements. The same is true for the case of 
impact. When impact takes place between two or more bodies, 
a certain amount of kinetic energy is always converted into heat. 
But it is absolutely impossible, by the direct application of heat, 
to restore the kinetic energy which was destroyed during impact. 
Also, when thermal equilibrium is established by mixing sub- 
stances, initially at different temperatures, the process is abso- 
lutely irreversible. 

29. Dissociation. Under Art. 26 we discussed the physical 
history of water with the continued application of heat up to the 
point of its superheated vapor. If heat be still further applied 
to the superheated vapor its temperature will continue to rise, 
up to some very high temperature, when complete dissociation 
takes place; i.e., the vapor splits up into its two constituent 
elements, viz, hydrogen and oxygen. Such a change is called a 
chemical change. If, while the pressure is maintained constant, 
heat be further applied, the gaseous mixture will rise in temper- 



PRODUCTION OF AND EFFECTS OF HEAT 29 

ature and increase in volume progressively with continued appli- 
cation of heat. If now, the process be reversed, i.e., the mixture 
be cooled, the temperature and volume will diminish until the 
temperature of dissociation is reached. When this point is reached 
the two gases recombine to form steam, and precisely the same 
amount of heat is evolved as was absorbed to bring about decom- 
position. The quantities of heat, however, which are involved 
in chemical decomposition and recomposition are very large in 
comparision with those quantities involved during changes in 
temperature and changes of physical state. Indeed, our greatest 
source of supply of energy, in the form of heat, is that due to 
chemical combination; viz, the combination of carbon, in the form 
of coal, with oxygen. 

It is true that the amount of dissociation is a function of the 
temperature; i.e., even water at ordinary temperatures has a 
small percentage of dissociation. This, however, does not 
invalidate the statement that, the energy absorbed during dis- 
sociation is equal to that liberated upon recombination. 

To give an illustration of the quantities of heat evolved during 
chemical combination, we may take as examples the combination 
of hydrogen and oxygen to form steam, and the combination of 
carbon and oxygen to form carbon-dioxide. In the former case, 
1 gram of hydrogen combining with oxygen to form steam 
(H2O), about 34,000 gram calories are evolved, or expressed in 
mechanical units, 1.43 X10 12 ergs. In the latter case, i.e., when 
1 gram of carbon combines with oxygen *to form carbon-dioxide 
(CO2), about 8000 gram calories are evolved, or, expressed in 
mechanical units, 3.36 X10 11 ergs. 

30. Electrolysis. Dissociation may also, in general, be brought 
about by electrolysis. That is, if an electric current be passed 
through a chemical compound, in the form of a solution, the com- 
pound will be split up into its constituents. 

Suppose that we are dealing with a solution of copper sulphate 
(CUSO4), and that the two electrodes are absolutely inert as regards 



30 HEAT 

chemical reactions. Then if an electric current be passed through 
the solution, copper will be deposited on the negative electrode 
(cathode), and the radical SO4 will be liberated at the positive 
electrode (anode). The SO4 thus liberated will combine with 
hydrogen of the solvent to form sulphuric acid (H2SO4) ; and at 
the same time oxygen will be liberated. The equation, repre- 
senting this reaction, is 

CuS0 4 +H 2 = H 2 S04+Cu+0. 

If an electric current be passed through water, then the water 
will be split up into its two elements, hydrogen and oxygen; 
hydrogen being given off at the cathode and oxygen at the anode. 

In the case of dissociation by heat, a definite quantity of heat 
disappears for a given amount of dissociation; and an evolution 
of an equal quantity of heat upon recombination. But, since 
a given quantity of heat represents a definite amount of energy, 
it follows that dissociation involves storing of energy. Likewise, 
when dissociation is brought about by electrolysis a definite 
amount of energy is consumed for a given amount of dissociation, 
which must necessarily be equal to that consumed when the same 
amount of dissociation is brought about by the application of 
heat; since the energy stored is the same in amount for both 
cases. It is true that a solution becomes heated when conveying 
a current; but, this has nothing to do with the dissociation. The 
development of heat being merely due to the resistance of the 
solution, the same as when any other conductor is conveying a 
current. 

It is immaterial, so far as the foregoing argument is concerned, 
whether we consider the solution initially partly ionized and the 
current merely a icarrier of the ions, or that the current actually 
splits up the compound. 

31. Faraday's Discoveries. Faraday showed experimentally 
that the amount of dissociation is directly proportional to the 



PKODUCTION OF AND EFFECTS OF HEAT 31 

time and the intensity of the current; and furthermore, that the 
amount of chemical action is the same for all parts of the circuit. 
The latter part may perhaps be best illustrated as follows : Assume 
that there are two voltameters connected in series; the first 
containing a solution of copper-sulphate and the second water. 
Then upon passing a steady current through the circuit a definite 
amount of copper will be deposited on the cathode of the first 
voltameter, for a given interval of time, and a definite amount of 
hydrogen liberated at the cathode of the second voltameter, 
during the same interval of time; and these two quantities will be 
in the same ratio as their chemical combining numbers. That is, 
for every gram of hydrogen set free at the cathode of the second 
voltameter, 31.59 grams of copper will be deposited on the 
cathode of the first voltameter; where, if hydrogen be taken as 
unity, 31.59 is the chemical equivalent of copper in copper-sul- 
phate. During the same time that the 31.59 grams of copper 
are being deposited on the cathode of the first voltameter, 1 
gram of hydrogen must be liberated to combine with the sulphion 
(SO4), set free to form H2SO4. 

32. Counter Electromotive Force. Since the amount of dis- 
sociation, other things being equal, varies directly as the cur- 
rent, and the amount of energy stored during dissociation depends 
upon the compound dissociated, it follows that every compound 
offers a definite counter e.m.f. to dissociation. And any applied 
e.m.f. less than this cannot bring about dissociation. To make 
this clear, the e.m.f. necessary to dissociate water will here be cal- 
culated. The amount of hydrogen set free, per coulomb of elec- 
tricity conveyed, is 0.000010357 grams. The number 0.000010357 
is called the electro chemical equivalent of hydrogen. 

Let, in the c.g.s. system of units, z be the electro chemical 
equivalent of hydrogen, I the current, and t the time, then the 
mass of hydrogen liberated, during the time t, is 

m = Izt . . (1) 



32 HEAT 

Let h be the heat, expressed in mechanical units, required to 
dissociate 1 gram of hydrogen, then 

mh = IEt; (2) 

where E is the applied e.m.f. Substituting in equation (2), the 
value of m as given in equation (1), we find 

Izth = IEt; 
from which 

zh = E. .. . . (3) 

If now, in equation (3), we substitute for z and h their values, 
remembering that for 1 gram of hydrogen, combining with oxygen 
to form steam, h = l.43Xl0 12 ergs; and that z, in the c.g.s. system 
of units, equals 0.00010357 grams per unit quantity of electricity, 
we find 



or, 



E = 1.43 X10 12 X 0.00010357 c.g.s. units e.m.f., 



w 1.43X10 12 X0.00010357 , ._ ., 

E = -^ = 1.48 volts. 

10 8 



It must, however, be emphasized that, in general, the e.m.f. 
for most cells is a function of the temperature; and, therefore, 
to calculate the e.m.f. this function must be known. We are not 
prepared, here, to take up this matter. The counter e.m.f. is 
readily determined by experiment. 

33. Junction of Dissimilar Metals. If the junction of two 
dissimilar metals be heated an e.m.f. is developed which is a func- 
tion of the temperature, and the metals which form the junction. 
To take a concrete example, assume a junction of antimony and 
bismuth. Such a junction, if heat be applied to it, develops an 
e.m.f. which tends to send a current from bismuth to antimony; 
and if a current be sent from bismuth to antimony, by the applica- 



PRODUCTION OF AND EFFECTS OF HEAT 33 

tion of an external e.m.f., there will be a tendency to reduce the 
temperature of the junction. On the other hand, if a current be 
sent through the junction from' antimony to bismuth, heat will 
be developed. 

34. As a resume, we may then state that the most general 
effects of heat are: To change the volumes of bodies; to bring 
about physical changes of state; in general, to promote chemical 
dissociation; to develop an e.m.f. at the junction of dissimilar 
metals. 

For the production of heat we may state the following examples : 
The production of heat by the mechanical compression of bodies 
which expand upon the application of heat; when the physical 
state of a body changes in the reverse order from the change when 
heat is applied; in general, by chemical combination; at the junc- 
tion of two dissimilar metals when a current is passed in a direc- 
tion opposite to that of the developed e.m.f. when heat is applied. 
Also, the production of heat when an electric current is conveyed 
by a homogeneous conductor; and, in general, when friction is 
being overcome, and mechanical motion destroyed by impact. 

35. The Principle of Energy. The various relations discussed 
in this chapter may now be summed up and stated quantitatively 
in a very simple manner. This generalization, known as the 
principle of energy, or conservation of energy, is one of the most 
extensive of generalizations, and may be stated in substance as 
follows: If in any system, from which no energy escapes and into 
which no energy enters, account be taken of all forms of energy, 
then no matter what transformations take place within the system, 
the sum total is a constant quantity. So far as experience goes, 
the foregoing statement is consistent with all phenomena; and 
hence, in all subsequent demonstrations its truth will be assumed. 



CHAPTER IV 
EXPANSION OF SOLIDS AND LIQUIDS 

36. Linear Expansion. It has been previously stated that, 
in general, bodies expand when the temperature is augmented. 
It is found by experiment that a body, such as a metal rod, 
increases in length by approximately equal amounts, between 
0°C. and 100°C, for equal increments of temperature. But, 
even though there is an approximate proportionality between 
change in length and change in temperature for moderate ranges, 
such as just specified, it must not be inferred that this is generally 
true for large ranges or high temperatures. 

If a body of unit length expand in length by an amount a 
for unit increment of temperature, then a body of length I will 
expand in length by an amount la for an increment of 1 degree; 
and for an increment of t°, between 0°C. and 100°C, the body 
will expand in length, approximately, by an amount Ion. Hence, 
if the length of the body at 0° be denoted by Zo, and at t° by l T , 
we have 

£ z = £o+ZoaT; 
from which 

Z t = Zo(1 + «t) (1) 

The quantity a is called the coefficient of linear expansion, and 
may be defined as the ratio of the change in length, per unit 
change in temperature, to the length at zero. The quantity in 
the parenthesis, viz, 1 + or, is called the factor of linear expansion. 

34 



EXPANSION OF SOLIDS AND LIQUIDS 35 

37. Voluminal Expansion. Homogeneous isotropic bodies 
will change in amount by like fractional parts of their original 
dimensions in all directions when the temperature changes. 
Assume that we are dealing with a rectangular parallelopiped 
whose three edges at zero temperature are ao, bo, and Co, its volume 
then, at zero, is 

vo = aob co (2) 

If the temperature, now, be changed to t°, the three edges become: 
ao(l + <XT), bo(l-\-ocz) } and co(l + aT); from which we have, for the 
volume at t°, 

v x = a boCo(l + on) 3 (3) 

Substituting in equation (3), for aoboco the value vo, as given by 
equation (2) , we have 

v T = v (l + Gn) 3 (4) 

Expanding equation (4), we find 

?; T = yo(l+3aT+3aV + a 3 T 3 ) (5) 

Now, a is a very small quantity in comparison with the dimen- 
sions of most bodies; hence, the two terms containing a 2 and a 3 
may be neglected; and we have 

r t =«b(l+3(rt) (6) 

Equation (6) is, of course, only approximately true; but, that it is 
correct for most practical purposes will be evident from an 
inspection of the following table, which gives the coefficients of 
linear expansion, per degree centigrade, for a few solids. It must 
be remembered that for substances like brass and glass the num- 
bers are only approximate; and in any case the value found for 
the coefficient of expansion will depend somewhat upon the treat- 
ment to which the specimen was subjected in its preparation. 



36 HEAT 



Coefficients of Lineak Expansion 

Platinum 0.899X10" 5 

Copper. . > 1.678X10 -5 

Steel (annealed) 1.095XKT 5 

Zinc 2.918X10" 5 

Brass . .0.187X10" 4 

Glass 0.083X10 -4 

Invar (steel containing 36% nickel) 0.087 X10~ 5 

It is only necessary to substitute the values of a, as given in the 
foregoing table, in equation (5) and it becomes evident that 
equation (6) is approximately true. Hence, we may say that, the 
coefficient of voluminal expansion is practically equal to three 
times the coefficient of linear expansion. 

38. Non-Isotropic Bodies. There are certain bodies which 
have different physical properties in different directions. Such 
bodies are termed non-isotropic. A notable example is that of 
Iceland spar, in which it is found that the coefficient of linear 
expansion in one direction is 2.63 X10~ 5 ; whereas, in a direction 
normal to this it is found to be only 0.544 X 10 ~ 6 . 

It is also interesting to note that Iceland spar manifests 
different optical properties in different directions. 

39. Expansion of Liquids. Liquids, in general, change more 
rapidly in volume than do solids for equal changes in temperature. 
However, since, in the case of liquids, the term linear expansion 
is meaningless, we deal only with voluminal expansion. 

The determination of the coefficient of linear expansion is 
quite simple; hence, the description of the methods employed in 
its determination was omitted. The determination of the coef- 
ficient of voluminal expansion of a liquid is attended by various 
difficulties; and, since the discussion of the principles involved will 
prove instructive, a few methods will be described. 



EXPANSION OF SOLIDS AND LIQUIDS 37 

Assume that we have a glass flask terminating in a tube of 
capillary bore, and that the mass of the flask when empty is known. 
The flask is then filled, to a definite mark on the capillary tube, 
with the liquid at a temperature ti, whose coefficient of voluminal 
expansion is sought. The mass of the vessel and contents is now 
determined. The difference between this mass and the mass of 
the flask gives us the mass of the liquid at the temperature ti. 
If Di is the density of the liquid at the temperature ti, and V 
the volume of the liquid in the flask at the same temperature, 
then 

D^f; (7) 

where M i is the mass of the liquid in the flask at the temperature 
ti. Let, now, M 2 be the mass of the liquid in the flask, when its 
temperature is T2, the flask being filled to precisely the same 
mark on the tube as when the temperature was ti. The volume 
of the flask will now be 

r = F{l + WT 2 -Ti)|; 

where @ is the coefficient of voluminal expansion of the glass of 
which the flask is composed. @ may be computed from the coef- 
ficient of linear expansion, or determined directly by experiment, 
as will be shown subsequently. The density of the liquid at T2, 
will now be 

M 2= M 2 m 

U2 V Fil + P(T2-Tl)} K) 

Dividing equation (7) by equation (8), we find 

g-gu+tt"-o)»- •••••• w 



38 HEAT 

If the coefficient of voluminal expansion of the liquid, between 
0° and the other two temperatures under consideration, is approx- 
imately constant, we have 

ft-.^. ... (10) 

where Do is the density of the liquid at 0° and a the coefficient of 
voluminal expansion. Similarly, we find 

D 2 =j^; (11) 

l+aT2 

from which, by dividing equation (10) by equation (11), we find 

D\ 1 + 0CT2 



Z>2 1 + GCTl 



(12) 



Finally, by equating the right-hand members of equations (12) 
and (9), we find 

1 + «T2 Mi 



1 + «T1 M 2 



1 + &(t 2 -ti)}. .... (13) 



In equation (13), all quantities excepting a are known; hence, 
its value is determinate. 

Another method is as follows: A solid, whose coefficient of 
voluminal expansion is accurately known, and which does not react 
chemically with the liquid whose coefficient of voluminal expansion 
is sought, is weighed in the liquid, first at temperature ti, and 
second at temperature T2. The weight of the solid being known, 
its loss of weight for the two temperatures is known; and, since 
the volume of the solid for any temperature may, from its known 
coefficient of voluminal expansion, be computed, the densities 
of the liquid for the two temperatures ti and T2 are readily found, 
and from these, as previously shown, the coefficient of voluminal 
expansion. 



EXPANSION OF SOLIDS AND LIQUIDS 



39 



I" 



w 



tt 



40. Direct Measurement of Coefficient of Voluminal Expan- 
sion. The method about to be described, and the one by which 
Regnault determined the coefficient of expansion of mercury, 
depends upon the principle that when communicating columns 
of liquids are in equilibrium their heights are inversely as their 
densities. In Fig. 3, AB and CD are two vertical iron tubes, 
cross connected by the horizontal tube BD. The two horizontal 
tubes, AE and CF, terminate in the vertical tubes EG and FI, 
which are connected by the inverted glass U-tube GJL The 
tube containing the stop cock s, 
is connected to the receiver of a 
compression pump; and after the 
apparatus has been filled with 
mercury, air is forced in from 
the compressor until the mercury 
in the tubes EG and FI is at 
a convenient height. The stop 
cock s is then closed. If, now, the 
temperature, and consequently 
the density, of the mercury in the 
tubes AB and CD be the same, 
the columns in EG and FI will be 
at the same level. On the other 

hand, if the temperatures of the two columns AB and CD be not 
the same, then the columns in EG and FI will not be at the same 
level. For, since there is free communication between B and 
D, the pressure must be the same at these two points. Further- 
more, the pressure of the air inside the U-tube being everywhere the 
same, and, since this pressure plus the pressure, due to the column 
in FI, balances the pressure due to the column DC, and the same 
pressure plus the pressure, due to the column in EG, balances that 
of the column BA, it follows that if the pressures, due to the two 
columns AB and CD, are not the same, the two columns in EG and 
FI cannot be at the same level. 



i__. 



E 

Fig. 3. 



40 HEAT 

Suppose, now, that the column in EG, FI, and CD be main- 
tained at 0°C. and the column AB at t°C, then the column in FI 
must exceed the column in EG, in height, by an amount /i,such that 

(H-h)(l + aT)=H; (14) 

where a is the coefficient of volummal expansion of mercury. 
From equation (14) we find 

(15) 



i(H-h) 



41. If, now, it be desired to determine the coefficient of expan- 
sion of some other liquid, it becomes only necessary to take a 
flask and determine its mass when empty, then the mass of flask 
and contents when filled to a given mark at various temperatures, 
first with the liquid whose coefficient is sought, and then when 
filled with mercury to the same mark for the various temperatures. 
From the mass of mercury required to fill the flask at various 
temperatures, and from the known density of mercury for these 
temperatures, the volume of the flask is readily computed. From 
the known volume of the flask and the mass of liquid required to 
fill it at the various temperatures, the densities corresponding 
to those temperatures are found. 



CHAPTER V 
'FUNDAMENTAL. EQUATIONS OF GASES 

42. Isothermal Equation. Experiment shows that, between 
certain limits, for the so-called permanent gases, such as hydrogen, 
oxygen, nitrogen, etc., the product of pressure and volume is 
a constant, for constant temperature. Expressed symbolically 

pv = k; ........ (1) 

where p is the applied pressure per unit area, * v the corresponding 
volume of the gas, and k some constant whose value depends 
upon the units chosen. Equation (1) is usually designated as 
Boyle's Law. But, in any case, the equation which expresses the 
relation between the pressure and volume of a gas, at constant 
temperature, is its isothermal equation. 

43. Gay-Lussac's Law. As a further result of experiment 
it is found that all gases which obey Boyle's Law have the same 
constant temperature coefficient; i.e.,' all gases, under constant 
pressure, expand by the same fractional part of their volumes at 
zero temperature, for equal increments of temperature. This 
is known as Gay-Lussac's Law. If we denote by a the increment 
in volume, for a unit volume of a gas, under a constant pressure 
po, when its temperature changes from zero to unity, then the 
volume of a gas at t degrees is 

v x = v +voon; (2) 

*In all subsequent equations, unless otherwise stated, p will be used 
to denote pressure per unit area. 

41 



42 HEAT 

where vo is the volume of the gas at zero temperature, and v x 
the volume,, under the same pressure, at t degrees. Writing 
equation (2) in another form, we have 

z; t =^o(1 + gct) . (3) 

If, after the temperature t, and the corresponding volume 
v z , under the constant pressure po, has been attained, the pressure 
is augmented, the temperature being maintained constant, until 
the gas assumes its original volume vo, we must, from Boyle's 
law, have the following relation: 

p x = p (l + on) (4) 

As a matter of fact the experiment is most conveniently per- 
formed, by varying the pressure, so as to maintain the volume 
constant, as the temperature is varied. Equation (4) may there- 
fore be considered as being the expression of experimental results. 
Multiplying both sides of equation (4), by vo, we obtain 

p T vo = p vo(l-\-(xx) (5) 

If, now, while the temperature is maintained constant, the pressure 
be varied, the volume will vary according to Boyle's Law; i.e., 

pv = p x vo = p vo(l+cn); (6) 

where p is any pressure and v the corresponding volume at the 
temperature t. 

Since Gay-Lussac' s Law also holds for temperatures below 
zero, a, of course, being a decrement, we have 

pv = poV (l — oct) (7) 

Equation (7) reduces to zero when <zt equals unity; hence 
t = l/a is the temperature below zero for which pv = 0. 



FUNDAMENTAL EQUATIONS OF GASES 43 

For the centigrade scale a = 0.003665, very nearly; hence, 
t = 1/0.003665 = 273, very nearly. Therefore, if there were no 
deviation from the relation expressed by equation (7), then at 
— 273 °C. the product of pressure and volume would become 
zero. As a matter of fact, all gases liquefy at temperatures above 
-273°C. This point, 273°C. below zero, is called the "absolute 
zero "; and temperatures measured from this zero are called tem- 
peratures on the "absolute scale." 

Substituting now, in equation (6), for a its value, viz, 1/273, 
we have 

p»=po»o(l+^)=fg(273+T). ... (8) 

But, in equation (8), po^o/273 is a constant for any particular mass 
of a gas, and 273 +t is the temperature as measured on the 
" absolute scale." Replacing the former by R and the latter by 
T, equation (8) becomes 

pv = RT (9) 

The numerical value of R depends, of course, upon the units 
chosen. 

Equation (9) is called the characteristic equation of a gas, and 
shows that the product of pressure and volume is directly propor- 
tional to the temperature as measured on the " absolute scale." * 

44. Departure from Boyle's Law. For ordinary pressures, 
Boyle's law is approximately true; i.e., for air the ratio of the 
product of pressure and volume, when the pressure is one atmos- 
phere, to the product of pressure and volume, when the pressure 
is two atmospheres, is about 1.002. The ratio of the initial value 
of pv to the final value of pv becomes greater as the final pressure 
becomes greater, until certain very high pressures, which are dif- 
ferent for the various gases, are reached ; after which the value 

* Since all knowledge is relative, the expressions "absolute zero" and 
"absolute scale" are not well chosen. 



44 HEAT 

of pv increases rapidly with increase of pressure. The values of 
the pressures for which, at ordinary temperature, the product 
pv is a minimum, are as follows : 100 meters of mercury for oxygen, 
65 meters for air, and 50 meters for both nitrogen and carbon- 
dioxide. For hydrogen the deviations from Boyle' 's Law are less 
and in the opposite direction. 

Since Boyle's Law is not rigidly true, it follows that all equations, 
which have been based on it, are not rigidly true. However, 
for ordinary ranges of pressure and temperature the character- 
istic equation is very nearly true. And, for the mathematical 
discussion of gases, it is very convenient to assume that we are 
dealing with a gas for which the characteristic equation is rigidly 
true. Such a gas is called a perfect or ideal gas. 

45. Gas Thermometer.* If a gas be confined in such a 
manner that its volume is maintained constant while the tem- 
perature varies, then it follows, from equation (9), that between 
those limits for which the equation is approximately true, the 
pressure varies directly as the temperature. All that is necessary 
then, to enable us to measure temperatures accurately, is a glass 
bulb of convenient size, filled with a gas, the most convenient being 
dry air, and some device by means of which the pressure can be 
regulated and measured. Such an arrangement constitutes a 
gas thermometer. Fig. 4 is a diagrammatic representation of 
the arrangement. A is a glass bulb, filled with gas, and connected 
by an inverted capillary U-tube, to the U-tube abc. B is a vessel, 
open at the top, partially filled with mercury, and communi- 
cates, by means of a flexible tubing, with the U-tube at b. The 
tube be being open at the top, the mercury in B and be must be 
at the same level. Assume that, when the temperature of A is 
at zero, the mercury in B, in be, and in ba is at the same level; 
then, if the temperature of the bulb A, and consequently that of 

* In all subsequent demonstrations, unless otherwise specified, the 
symbol T will be employed to designate temperatures as measured by 
the ideal gas thermometer, the zero of which is about — 273 °C. 



FUNDAMENTAL EQUATIONS OF GASES 



45 



^ 



O 




the gas in it, be increased, the gas will expand if the pressure be 
constant. Hence, to maintain the gas at its original volume, 
the vessel B must be raised so as to increase the column be, in a 
manner, such that the column ba, and hence the volume of the gas. 
is maintained constant.' The 
difference in the height of the 
columns be and ba measures 
the increment in pressure; and 
hence, the increment in tem- 
perature may be calculated. A 
correction must, however, be 
made for the change in volume, 
due to change in temperature, 
for the bulb A. This correction 
is readily applied, provided the 
coefficient of voluminal expan- 
sion for the glass, confining the 
gas, be known. 

Let po and vo, respectively, rep- 
resent the pressure and volume 

of the gas for the temperature at zero, p v the observed pressure 
for the temperature t, a the coefficient of expansion for the gas, 
and {J the coefficient of voluminal expansion for the glass. The 
new volume, then, for the gas at the temperature t, will be ?>o(l + (k) ', 
hence 

p t «o(l+(fc)=;po«D(l+aT). 




Fig. 4. 



Now, p T = po(l + 2''t); where a' is the apparent coefficient of 
expansion of the gas. Making this substitution, for p T , and 
eliminating, we find 

(l+«'T)(l+PT) = l+arc; 



from which 



T=- 



«- («'+£) 
«'(* 



46 HEAT 

When extreme precision is sought, corrections must also be 
made for changes in volume of the bulb, due to changes in pressure. 
Gas thermometers are used only for purposes of standardization. 

46. Expansion without Doing External Work. Experiment 
has shown that there is no energy consumed in the simple expan- 
sion of a gas; i.e., when a gas expands in such a manner that no 
external pressure is overcome, and hence, no external work is 
being done, no energy is consumed by it. This experiment was 
first performed by Gay-Lussac (who apparently did not realize 
its full significance) in the following manner: Two vessels, one 
of which was exhausted and the other filled with air under a pres- 
sure, were placed in a calorimeter and surrounded by water. 
When the stop cock in the tube, connecting the two vessels, was 
opened, the air from the one vessel expanded into the other, 
bringing about an equalization of pressures. During this process 
the gas increased in volume without doing any work external 
to the system. That is, work was done by the gas under a high 
pressure, in the one vessel, in expanding against the increasing 
pressure of the gas in the other vessel. But, since the temperature 
of the system after the completion of the process was found to 
be precisely equal to that of the system before the process ^began, 
it follows that the total energy of the system is unchanged by the 
expansion. If energy were required to bring about an increase 
in volume, the temperature of the system at the end of the process 
would necessarily be less than at the beginning of the process. 
Or, to express the result in another way, since the temperature of 
the system is unchanged by the change in volume, the work done 
by the gas in the vessel, initially under the higher pressure, is pre- 
cisely equal in amount to the work done on the gas in the other 
vessel. The results just deduced may be embodied in a simple 
statement; i.e., the intrinsic energy of a perfect gas is a function 
only of the temperature. Or, to put it still in another way, the 
heat of disgregation of a perfect gas, when no external work is 
being done, is zero. 



FUNDAMENTAL EQUATIONS OF GASES 47 

The foregoing experiment was subsequently repeated, in the 
most careful manner, by Joule and found to be approximately, 
though not rigidly, true. 

47. Thermal Capacities of Gases. If a gas, under a pressure 
p, expand by an amount in volume dv, the external work done will 
be numerically equal to pdv; where p is the pressure per unit 
area. This is shown as follows: From the definition of work, 
we have 

dw = Fds; (10) 

where F is the applied force and ds the displacement. But, since 
F, the applied force, is numerically equal to the product of p, 
the pressure per unit area, and the area A, we have 

dw = pAds (11) 

But 

Ads = dv; 

hence, by substituting in equation (11), we have 

dw = pdv (12) 

Since, according to the experiments of Gay-Lussac and Joule, 
when the temperature of a gas is augmented, that part of the heat 
which is required to elevate the temperature of the gas is practically 
the same for equal ranges of temperature, no matter whether the 
volume be varied or maintained constant, it follows that the quan- 
tity of heat required to bring about a given elevation of temper- 
ature, when the pressure is maintained constant, is greater than 
the quantity of heat required to bring about an equal elevation 
of temperature, when the volume is maintained constant. For, 
in the former case, heat is required, not only to elevate the temper- 
ature of the gas, but also to do the external work due to the expan- 
sion of the gas; whereas, in the latter case, the heat consumed is 
only that required to elevate the temperature of the gas. 



48 HEAT 

The ratio of change in heat to the corresponding change in 
temperature, in a unit mass of gas, when the volume is maintained 
constant, is a measure of the thermal capacity per unit mass at 
constant volume, and is denoted by Cv Hence, for a unit mass 
of gas, we have 



In a like manner 



ST 7 



(13) 



= C P ; (14) 



where C p represents, for a gas, the thermal capacity per unit mass, 
under a constant pressure. 

The ratio C p /C v = n is practically a constant for all the per- 
manent gases; and, in the case of air, is approximately 1.405. The 
quantity (Cp—C v ), is evidently a measure of the external work 
done by the unit mass of gas in expanding against the pressure 
p, while it is being heated through a range of 1 degree. 

Assume that we have a given mass of gas m, whose volume 
is v at 0°C, confined in a cylinder by a piston of area A, under a 
pressure p, per unit area. If the piston be perfectly free to move, 
and heat be applied bringing about an elevation of temperature 
t, the pressure being maintained constant during the process, 
then the volume will be increased by an amount von. The dis- 
tance through which the piston moves during the expansion is 
von /A; and the external work done, which is numerically equal 
to the product of force and displacement, is 



W = pA-~- = pvai: (15) 



The heat consumed in doing the external work, expressed in 
mechanical units, is 



H = Jm(C p -C v )'z; ........ (16) 



FUNDAMENTAL EQUATIONS OF GASES 49 

where J is the mechanical equivalent of heat. But, under the 
assumed conditions, H and W are numerically equal; hence, from 
equations (15) and (16), we have 

Jm(C p — C P )T = pyax; 
from which 

j- Wl (m 

Equation (17) * enables us to compute the mechanical equiv- 
alent of heat from the known constants of a gas. . The constants 
of dry air are as follows: C p = 0.2375, C„ = 0.1690, a = 0.003665, 
and the mass of 1 c.c. of air at 0°C, under a pressure of 1.01325 
X10 6 dynes per sq.cm., is 0.001293 grams. Substituting these 
values in equation (17), and assuming 1 c.c. for the initial volume, 
we find 

T 1.0132 X10 6 X 0.003665 . 1ft o vin7 , • 

J = 0.001293(0.2375-0.169 0) =4 - 193X1 ° ergS per gram Calone ' 

The value of J thus obtained differs by a small percentage from 
that obtained by the direct conversion of mechanical work into 
heat. We have here then a complete verification of the numer- 
ical relation between heat and work. That is, in the one case, 
mechanical work is directly converted into energy, in the form 
of heat, and the numerical ratio of the two quantities involved is 
determined. In the other case, heat is converted into work, and 
the numerical relation between the two quantities is again 
determined; and the difference, between the two values so deter- 
mined, is well within the limits of observational error. 

* It was by this method that J. R. Mayer first computed the mechanical 
equivalent of heat. Various writers have attempted to take some of the 
credit from Mayer, by asserting that it was not then known that no energy- 
is required for the simple expansion of a gas. But Gay-Lussac had per- 
formed this experiment, and anyone reading Mayer's original papers will 
see that he was aware of this and interpreted the experiment properly. 



50 HEAT 

48. Adiabatic Equation. If a gas is compressed, work is being 
done on it and heat is necessarily developed; and since the pres- 
sure of a gas, other things being equal, rises with the temperature, 
it follows that, unless the heat developed by the compression is 
abstracted from the gas as rapidly as it is developed, the pressure 
must rise more rapidly, with respect to the amount of compres- 
sion, than it would during isothermal compression. There are 
certain processes where compression and dilatation take place 
so rapidly that there are practically no exchanges of heat between 
the various parts of the system. Changes, during which no heat 
enters or escapes, are called adiabatic changes. 

A good example of adiabatic changes are the compressions 
and rarefactions which take place in a medium when a sound 
wave exists in it ; the time of compression, or rarefaction, being so 
small that practically no heat is transferred from particle to 
particle. 

In general, if we are dealing with a unit mass of a perfect gas, 
we may write 

dQ = C v dT+pdv; (18) 

where all quantities, of course, are expressed in the same units. 
dQ, expressed in mechanical units, is the quantity of heat absorbed 
by the gas, or else abstracted from it, C v dT is the quantity of heat 
involved in bringing about the change of temperature dT, and 
pdv represents work done either by the gas or on the gas. 

As a matter of illustration, assume work is being done on the 
gas in a manner such that its temperature rises and that there is 
also heat given to the surroundings. If we consider work done 
by the gas and heat absorbed by the gas positive, then, in equation 
(18), if applied to this case, both dQ and pdv become negative; 
on the other hand, C v dT remains positive. 

If, now, during the change in volume, no heat enters or escapes, 
the process will be adiabatic; and equation (18) becomes 

C v dT+pdv = (19) 



FUNDAMENTAL EQUATIONS OF GASES 51 

In equation (19), if dv is positive, dT must be negative; i.e., if 
external work is done by the gas it is done at the expense of the 
intrinsic energy of the gas, and the temperature must fall. Like- 
wise, if work is done on the gas, since according to our assumption 
no heat escapes, its temperature must rise. To solve equation 
(19) we will substitute for dT its value, as found from equation 
(9), Art. 43; i.e., by differentiating 

pv = RT, 
we find 

dT = pd»+vdp (20) 

Now, R can be expressed in terms of the two thermal capacities. 
To find this relation, assume that we are dealing with a unit mass 
of the gas and allow it to expand under a constant pressure p, 
while its temperature is increased by unity. If v\ is the initial 
and V2 the final volume, the external work done is 

p(V2 — Vi)=Cp — Cv. 

It being understood that all quantities are measured in mechan- 
ical units. But, from equation (9), it follows directly, that 

p(v 2 -v 1 )=R(T 2 -T 1 ); 

and, since by the conditions we have T 2 — T\ equal to unity, it 
follows that 

it = O p Op. 

Substituting this value of R in equation (20), and the value of 
dT so obtained in equation (19), we find 

^pdv+vdp 
Cv C -C +P dv = > 
from which 

/Cp\ dv dp 



C> v + J = °- ....... TO 



52 HEAT 

But, as previously explained, C v /C v is practically a constant. 

Representing this constant by n and substituting in equation 

(21), we have 

do .dp 
n — h— = 0; 
v p 

from which, by integration, 

\ogv n -\-\ogp = ki; 

where k\ is the constant of integration. Or, expressed in another 
form, 

log po n = ki; 
from which 

po n = k; (22) 

where k is a constant depending upon the units chosen. Equation 
(22) gives the relation of pressure and volume for a gas, during 
adiabatic changes, and is known as the adidbatic equation. 

General Equations of Gases 

49. The change of heat involved when a gas suffers a change 
is a function of the temperature, pressure, and volume; i.e., 

Q=f(T,p,v). 

But, since any two of these quantities may vary independently 
of the third, we may write 

Q-f(T,p), Q=f'(T,v), and Q=f"(p,v). 

By partial differentiation of these functions , we obtain 



and 



dQ-C 



<!)*+(£)*■■ (25) 



FUNDAMENTAL EQUATIONS OF GASES 53 

If it be assumed that we are dealing with a unit mass of gas, 
then in equation (23), we have 

(§)dT=C,dT; 

since in this case, ^ is the thermal capacity per unit mass 
at constant pressure. In a like manner, 

( 



m 



is also a unit thermal capacity, and is the ratio of change in heat 
to change in pressure at constant temperature. Substituting, 
in equation (23), we have 

dQ = C p dT+mdp (26) 

In equation (24) 

(§)dT=CJT; 

since (ttp) * s the thermal capacity per unit mass of the gas at 
constant volume. In a like manner, 



m 



= 1 

T 



is a unit thermal capacity, and is the ratio of change in heat to 
change in volume at constant temperature; hence, by substi- 
tuting in equation (24), we find 

dQ = C v dT+ldv (27) 



Again, in equation (25) the two quantities, viz, ( — J , the ratio 

of change in heat to change in pressure at constant volume, and 

/oQ\ 

( ^— J j the ratio of change in heat to change in volume at constant 



54 HEAT 

pressure, are both unit thermal capacities. If we represent the 
former by j, and the latter by o, then equation (25) becomes 

dQ=jdp+odv. (28) 

Since all of these equations must be true for the particular 
case when the left-hand members are equal, the right-hand mem- 
bers of equations (26) and (27) may be equated, and we have 

C p dT+mdp = C v dT+ldv (29) 

From the fundamental statement 

v=f(p, T), 



we have 



a*+ 



-(|^1 dT 



Substituting this value of dv, in equation (29), we have 
C p dT+mdp = C v dT+l(^Jtp+l(^dT; 



and 

C p dT+mdp 



1 ■ : ^),} dT+l (%\ dp - ■ ■ (30) 



Since this equation is true when the corresponding changes 
in the two members are equal, it follows that 



and 

Again 
from which 



>. '■->■(%) 






(fWS>° 



FUNDAMENTAL EQUATIONS OF GASES 55 

Substituting the value of dp as given in equation (33), in equation 
(29), we have 

CpdT+m [ (£)*+(!£) W\ =C„dT+ldv; 
from which 

hence, by equating like coefficients, we find 

C p -C,= -m(||) o (34) 

By equating the right-hand members of equations (26) and (28) , 
we find 

C p dT+mdp=jdp+odv (35) 

From the fundamental statement 

T=f(v, p), 



we have 

dT=( 

and substituting this value of dT, in equation (35), we obtain 



-) dv+(^)dp; 



| /8T\ 



Cp J \to) p dv+ X^) v dv I +mdp =J d V+odv, 
hence, by equating like coefficients, we find 

= ^(£)/ ■•-.... 06) 
From equations (27) and (28), we have 

CvdT+ldv =jdp + odv ; 



56 HEAT 

and, by substituting for dT its value, we find 

from which 

*-o. <*> 

We then have the following values : 



: (sr), 



l\ ^m) Co L>V) 



l[h £)r m > 



»Mg|) =-(.G,-C.), 



and 



1 if > - 



^ - 



From the characteristic equation 

pv = RT, 
we find 



hence 



Again 



hence 



'iv\ R 




lJLrr _m 




l -pK^P ^vj» • • • 

lv\ RT 

IvJt p 2 '' 




T 

m= (C p — C V )o o e 

p 


o . . 



(38) 



(39) 



FUNDAMENTAL EQUATIONS OF GASES 57 

Also 



(— \ = -• 



hence 



j=^fi, (40) 



Finally 

\lv] v R' 
and 



o = \Cv (41) 



Substituting in equations (26), (27), and (28) the values of I, m, 
j, and o as just determined, we find 

dQ = C p dT--(C P -C v )dp, (42) 



dQ = C v dT+^(C p -C v )dv, (43) 



and 



dQ = ^C v dp+^C p dv (44) 



These equations, viz, (42), (43), and (44), may be put into 
different forms; since, from the characteristic equation 

pv = RT, 
we have 

T/p = v/R, and p/R = T/v. 

From the assumption, then, that in the fundamental statement 

Q=f(T,p,v), 



58 HEAT 

any two of the variables may vary independently, while the third 
is maintained constant, we have obtained three distinct equa- 
tions, viz, equations (42), (43), and (44). It is of further interest 
to note that if in any one of these three equations the right-hand 
member is equated to zero, the adiabatic equation is obtained. 

Assuming that the change in equation (44) is adiabatic, then 
dQ = 0, and we have 

C v vdp = —C p pdv; 



from which 



and 



from which 



C p dv _ _dp 
fdv (dp 



n\ogv= — logp+fci. 

Or, expressed in another form 

log pv n = k\ ; 
and, finally 

pv n = h (45) 

This may also be obtained from the other equations, as may be 
readily shown. From 

pv = RT, 
we have 

dT= pdv±vd2 
R 

Substituting this value of dT in equation (43) and equating to 
zero, we find 

C v (pdv-\-vdp) = —p(C P — Cv)dv; 
from which 

C v vdp= —C v pdv; 



FUNDAMENTAL EQUATIONS OF GASES 59 

which is identical with the result obtained from equation (44) 
under the same assumption. Again, substituting in equation 
(42), for dT its value and equating to zero, we find 



from which 



c vMp^ {Cp _ Ct)dp . 



C p pdv = —Cvvdp; 



which is again identical with that previously obtained under the 
same assumption. 

If it be desired to find the temperature of a gas, corresponding 
to a given pressure and volume, during an adiabatic change, in 
terms of the initial temperature and pressure and the given pres- 
sure, or in terms of the initial temperature and volume and the 
given volume, we proceed as follows: Let T\, pi, and v\ be, re- 
spectively, the initial temperature, pressure, and volume, p and 
v, respectively, the pressure and volume, for which the corre- 
sponding temperature, T, is sought. Then, since the two points 
are on the same adiabatic, we have 

pivi n = pv n ; (46) 

and from the characteristic equation, we have 

piv^RTu (47) 

and 

pv = RT (48) 

Substituting in equation (46) , the values of v\ and v, as found from 
equations (47) and (48), we find 



60 HEAT 

In a similar manner, by substituting in equation (46), the values 
of pi and p, as found from equations (47) and (48), we obtain 

T=Ti (j) n ~- (50) 



Vapors 

50. Vaporization. The gaseous states of bodies, which under 
ordinary conditions of temperature and pressure are either liquids 
or solids, are called vapors; and the process by which the vapor 
is formed is called vaporization. 

In general, vaporization takes place in two distinct ways. In 
the one process, called evaporation, vapor is continually formed 
at the exposed surfaces of liquids; and in the other process, called 
ebullition, bubbles of vapor are formed in the body of the liquid 
or at the heated surfaces. 

51. Evaporation. If a liquid be enclosed in a space, only 
part of which is occupied by the liquid, then vapor immediately 
forms and occupies the space above the liquid. This continues 
until the vapor has reached a certain density which depends 
upon the temperature, and is greater as the temperature is higher, 
but is always the same for the same temperature. In other 
words, for any given temperature there is a maximum density 
and hence, a maximum pressure, which the vapor is capable of 
exerting. When this state is reached the vapor is said to be 
saturated. That is, for the given temperature the space contains 
the maximum possible amount of vapor. If, after this state has 
been reached, the temperature be maintained constant and an 
attempt be made to increase the pressure by the application of 
an external force the result will be, not an increment in pressure, 
but a diminution in vdlume, at constant pressure, and a corre- 
sponding amount of condensation. In other words, the pressure 
for a saturated vapor at constant temperature is constant. Or, 



FUNDAMENTAL EQUATIONS OF GASES 61 

to put it in still another way, the temperature of a saturated 
vapor is uniquely defined by its pressure. 

52. Addition of Vapor Pressures. The rate of evaporation 
depends, of course, upon the rate at which heat is being supplied; 
but, as has just been stated, the final pressure reached depends 
merely upon the temperature. Furthermore, evaporation takes 
place more rapidly in a vacuum than in a space occupied by the 
vapor of some other substance ; however, the final pressure reached 
by the vapor will be almost, though not quite, as high, when the 
space is partially occupied by some other gas or vapor, than it 
would be were the space originally a vacuum, provided always, 
that the temperature be the same and that there be no chemical 
action between the vapors. This statement was first made by 
Dalton, viz, when evaporation takes place in a space filled by 
another gas, which has no action on the vapor, the final pressure 
reached by the mixture is equal to the sum of the pressures of its 
constituents. Careful experiment shows Dalton's statement to 
be approximately, though not rigidly true. 

53. Ebullition. As has been previously stated, when heat is 
applied to a liquid, the temperature rises progressively with 
continued application of heat until a certain point, which depends 
upon the pressure, is reached, when the temperature remains 
constant. This is the boiling-point for the given pressure; and 
is that temperature for which the pressure of the vapor is equal 
to the superimposed pressure. Since the pressure at any point 
in the liquid, is equal to the pressure at the surface plus the pres- 
sure due to the liquid, from the surface to the point under con- 
sideration, it follows that, the temperature varies for different 
depths below the surface of the liquid. Hence, the temperature 
of the boiling liquid is not a constant throughout; but increases 
slightly with the depth. 

When equilibrium has been attained, i.e., the temperature 
becomes constant, then all the energy that is supplied, in the form 
of heat, is consumed in converting the liquid into a vapor. This 



62 HEAT 

energy consists of two parts, viz, one part being that energy 
which is required to overcome the inherent forces, that is, to 
separate the particles so as to form vapor, and the other part 
to overcome the external pressure during the augmentation of 
volume. The former is called the heat of disgregation and the 
latter the heat of expansion. 

Pressure, however, is not the only factor that fixes the boiling- 
point of a liquid. As examples, the following may be cited: 
The nature of the material of the containing vessel has some influ- 
ence. If the liquid be first carefully freed from the imprisoned 
air, the temperature may be raised considerably above the tem- 
perature at which ebullition ordinarily takes place. Impurities 
in the liquid influence the boiling-point. And finally, salts dis- 
solved in ajiquid always raise the boiling-point. As an example, 
the boiling-point of a saturated solution of water with common 
salt is about 109°C. But the temperature of the saturated vapor 
of a liquid is always the same for the same pressure, no matter 
what the temperature of the liquid. It is for this reason that the 
temperature of steam, rather than that of water, under a pressure 
of one standard atmosphere has been chosen as the boiling-point. 

54. Critical Temperature. When a liquid is heated in a closed 
vessel the vapor accumulates above the liquid and augments 
the pressure. Up to a certain point, differing for different liquids, 
there is a sharp definition between the liquid and vapor; but, for 
every liquid there is reached, finally, a temperature when this 
definition ceases, and the liquid disappears and is completely 
converted into vapor, even though the volume occupied by the 
vapor is but little greater than that occupied by the liquid. The 
temperature at which this takes place is called the critical tem- 
perature for the substance. And, it appears that, for temperatures, 
higher than this, no matter what the applied pressure, the substance 
can exist only in the gaseous state. The following table gives a few 
substances together with their approximate critical temperatures, 
and the corresponding pressures: 



FUNDAMENTAL ^EQUATIONS OF GASES 



63 



Substance. 


Temperature in 
Degrees, C. 


Pressure in 
Atmospheres. 


Carbon-dioxide 

Sulphur-dioxide 


31 

156 

194 

365 

-118 

-146 

-234 


77 
79 
36 
195 
50 
33 
20 


Ether 


Water 


Oxvsen 


Nitrogen 


Hydrogen 



55. It is interesting to note that there apparently is a relation 
between heat of disgregation of a substance and its critical tem- 
perature. Let a be the specific volume of the liquid; i.e., the 
volume occupied by unit mass of the liquid, and s the specific 
volume of the dry saturated vapor, then the increment in volume, 
when a unit mass of a liquid is converted into vapor, is 



/j. = s-<s; 



(51) 



where fx is the increment in volume. The external work, or the 
heat of expansion, is 



W = pn = p(s-G); 



(52) 



where p is the pressure during the evaporation. To express the 
heat of expansion, during evaporation, in thermal units, we must 
divide by J, and equation (52) becomes 

W = Ap(s-v); (53) 

where A = l/J. 

If we represent by r the heat of vaporization, and by p the 
heat of disgregation, equation (53) becomes 



from which 



r-p = Ap(s-a); 

p = r-Ap(s-a). 



(54) 
(55) 



64 HEAT 

To illustrate, we will now compute the heat of expansion for 
water when it is converted into dry steam at 100°C. 

Assume that we are dealing with 1 gram of water at 100° 
C. Its volume will be approximately 1 c.c; and the volume of 
its saturated vapor under atmospheric pressure will be about 
1670 c.c. Substituting, in equation (54), for s — cr, A, and p the 
numerical values, we find 

1.013 X10 6 X 1670* ... . . 

r— p = iQtwio 7 = 40.3 gram calories per gram. 

This gives for the heat of disgregation, which is the difference 
between the heat of vaporization and the heat of expansion, 
536.5-40.3 = 496.2. 

Zeuner gives an empirical equation, for the heat of disgregation 
for water, which gives results very close to those obtained by 
equation (55) ; this equation is 

p = 575.4 -0.79 It; ...... (56) 

where p is in gram calories per gram and t in degrees centigrade. 

In general, the heat of disgregation becomes less as the tem- 
perature becomes higher; and the critical temperature appears 
to be that for which the heat of disgregation becomes zero. In 
the case of some liquids, very close agreement is found between the 
values for the critical temperatures, as found by direct experiment, 
and those values calculated from the empirical equations for the 
heat of disgregation. In other cases, again, there are discrepancies 
of considerable magnitude. A notable case is that of water; 
but, it must be remembered that the critical temperature of water 
is very high, and therefore, its determination is attended by dif- 

* s not being known definitely to the fourth significant figure, it is 
immaterial whether we use 1670 or 1669; or, in other words, the volume 
of the liquid, in this case, is negligibly small in comparison with that of its 
vapor. 



FUNDAMENTAL EQUATIONS OF GASES 65 

Acuities. Furthermore, the heat of vaporization for water has 
not been determined for such high temperatures; hence, the empir- 
ical equation for such ranges is doubtful. 

56. Total Heat of Steam. Before leaving the subject of satu- 
rated vapors, we will give, on account of the importance in steam 
calculations, the empirical equations for the total heat of satu- 
rated steam, and the heat of vaporization. By total heat of steam 
is meant the quantity of heat required to raise the temperature 
of unit mass of water, from the melting-point of ice, to the tem- 
perature under consideration and convert it into saturated steam 
at that temperature. In the French system the total heat is 
given by the equation 

H = 605 +0.305 t gram calories per gram. . . . (57) 

In the English system the total heat is given by the equation 

H = 1082+0.305 t B.T.U. per pound (58) 

In equation (57), H equals the quantity of heat, in gram calories, 
required to raise 1 gram of water from 0°C, to the temperature 
t°C, and convert it into a saturated vapor at that temperature. 
In equation (58) , H equals the quantity of heat, in British thermal 
units, required to raise 1 pound of water from 32°F. to t°F., 
and convert it into a saturated vapor at that temperature. 

57. Heat of Vaporization for Water. The empirical formula, 
which gives the heat of vaporization of water, for the various 
temperatures, is 

r = 1114-0.7T B.T.U. per pound. . . . (59) 

Equation (59) is not quite as accurate as equation (58); but 
for most steam calculations it is sufficiently precise; since, by 
means of it the heat of vaporization is found, with an error of less 
than 1 per cent, between 100°F. and 400°F. 



66 HEAT 

58. Superheated Vapors. Any vapor which, for a given 
pressure, is at a temperature higher than that corresponding to 
saturation, for the given pressure, is said to be superheated. This 
is only possible when the vapor is not in contact with its own 
liquid. Vapors which have been superheated obey Boyle's law 
approximately; and furthermore, for adiabatic changes, the 
equation 

pv n = k, 

holds; n having different values, depending upon the vapors 
with which we are dealing. 

59. Hygrometry. Hygrometry has for its object the deter- 
mination of the state of the atmosphere with respect to the aqueous 
vapor present. The amount of aqueous vapor present in the 
atmosphere is a very variable quantity. The effect of the vapor, 
however, depends not only upon the quantity present, but also 
upon the temperature. These two facts are included in the single 
statement, that the effect of the vapor depends upon the relative 
humidity. 

By the expression relative humidity, is meant the ratio of the 
actual density of the vapor, contained by the air, to the density 
which it would have if there were saturation for the given tem- 
perature. Or, if expressed as a percentage, the relative humidity 
is the percentage of saturation for the given temperature. 

60. Dew Point. The dew point is that temperature at which 
the vapor present in the atmosphere, begins to condense; i.e., 
the point of saturation. Assume that a certain portion of the 
atmosphere is cooled until the vapor present begins to condense. 
The temperature at which this takes place is readily found by 
experiment. By referring to a curve giving the relation of 
temperature and pressure of saturated steam, we can readily 
find the pressure corresponding to the dew point. Designating 
this pressure by pi, and by p2 the pressure of saturated vapor 
corresponding to the temperature of the atmosphere, then since 



FUNDAMENTAL EQUATIONS OF GASES 67 

non-saturated vapors approximately obey Boyle's law, the density 
of the vapor present in the atmosphere is to the density the vapor 
would have were there saturation, very nearly, as p\ is to P2. 
Hence, we have approximately, for the relative humidity 

h = 100^ per cent. 

To illustrate further, we will consider a concrete case. Assume 
the temperature of the atmosphere to be 25 °C, and that of the 
dew point 15°C. From the steam curve we find pi = 1.278 cm. 
of mercury, and p2 = 2.369 cm. of mercury. From this, the rela- 
tive humidity, expressed as a percentage, is found to be 



1 278 
ft = 100X^7^=53.9 per cent. 



61. Absolute Humidity. The amount of moisture, expressed 
in grams, contained by a cubic meter of air is called the absolute 
humidity. This is found in a very simple manner. The dew 
point is determined, giving the temperature of the saturated 
vapor, and from this, by referring to steam tables, the mass per 
unit volume is found. 



CHAPTER VI 
ELASTICITIES AND THERMAL CAPACITIES OF GASES 

62. The adiabatic equation, for gases and vapors, being so 
frequently employed in the discussion of the theory of heat 
motors, it is important to say a few words about the determination 
of the ratio C p /C v = n. The determination of thermal capacities 
of gases is attended by far greater difficulties, than those found 
in the determination of thermal capacities of liquids and solids. 
This is due to the fact that the thermal capacity of a gas is always 
very small in comparison with that of the containing vessel. 

63. Thermal Capacity at Constant Pressure. Regnault was 
the first to determine accurately the thermal capacities of gases 
under constant pressure.* The method is essentially as follows: 
The gas, whose thermal capacity is sought, is contained in a large 
reservoir, under a high pressure, from which it is passed through 
a spiral tube immersed in a bath, the temperature of which is 
maintained constant. The spiral tube is of sufficient length to 
insure the gas leaving it to be at the same temperature as the 
bath. In the tube, connecting the reservoir with the spiral tube, 
is a valve, by means of which the pressure of the gas is maintained 
constant. A second spiral tube, through which the gas must 
pass, is immersed in a calorimeter rilled with water; the thermal 
capacity of the calorimeter and contents being known. The spiral 
tube, immersed in the calorimeter, is of such length that the tem- 
perature of the gas, throughout the progress of the experiment, 
is reduced to that of the calorimeter before being discharged into 

* For a full description, see Preston's "The Theory of Heat," Chapter IV, 
Section VI. 



ELASTICITIES AND THERMAL CAPACITIES OF GASES 69 

the atmosphere. From the initial and final pressure of the gas 
in the reservoir, together with its temperature, which is main- 
tained constant, by means of a suitable bath, the mass of gas pass- 
ing through the calorimeter, during the progress of the experiment, 
is readily found. Furthermore, from the thermal capacity of 
the calorimeter and contents, together with the initial and final 
temperatures, the quantity of heat absorbed by the calorimeter 
during the progress of the experiment, proper corrections being 
made for losses, is determined. And, since the temperature of 
the gas before entering the calorimeter, as well as the average 
final temperature, is known, and also the mass of gas which has 
passed through the calorimeter, the thermal capacity per unit 
mass is determinate. 

64. Thermal Capacity at Constant Volume. In the experi- 
ment just described the quantity of gas employed is not limited 
by any containing vessel; for, the reservoir in which the gas is 
contained may be of any size whatsoever, without having any 
influence on the result. Therefore, a large quantity of gas may 
be used, and consequently, a considerable range of temperature 
may be obtained in the calorimeter. However, when it is desired 
to determine the thermal capacity of a gas at constant volume, 
the quantity of gas upon which we are experimenting, is limited 
by the containing vessel; and the thermal capacity of the con- 
taining vessel is always large in comparison with the thermal 
capacity of the enclosed gas. 

65. Joly's Steam Calorimeter.* In the most primitive form, 
the steam calorimeter consists of the pan of one side of a beam 
balance placed in an enclosure with the specimen, whose thermal 
capacity is sought, supported by the pan. When steam is admit- 
ted into the enclosure, condensation takes place until the temper- 
ature of the test specimen is equal to that of the steam. The 
steam then passes through the enclosure without further conden- 

* For a complete description of this apparatus, see Preston's "The Theory 
of Heat," Chapter IV, Section V. 



70 HEAT 

sation. When this condition has been reached, the balance is 
counterpoised and the mass of condensed steam, which has been 
collected by the pan, is noted. From the initial and final temper- 
ature of the test specimen, together with the quantity of water 
collected by the pan, and the heat of vaporization for this par- 
ticular temperature, the thermal capacity of the test specimen is 
readily found . 

66. Differential Steam Calorimeter. In this form, both pans 
of the balance, which are made so that they have equal thermal 
capacities, are suspended in the enclosure. On the one pan is 
placed a spherical vessel, which has been exhausted, and on the 
other pan a spherical vessel of like dimensions and equal thermal 
capacity, filled with the gas whose thermal capacity is sought. 
When steam is now admitted into the enclosure, the quantity of 
water which collects Jn the pan, supporting the vessel containing 
the gas, is greater than that which is collected in the pan supporting 
the exhausted vessel. This is necessarily so; since the vessel, 
together with the contained gas, has a thermal capacity greater 
than the exhausted vessel. From the excess of condensation in 
thejDne pan over that in the other, which is obtained directly 
by weighing, together with the initial and final temperatures 
of the enclosure and the mass of gas contained by the one vessel, 
the thermal capacity per unit mass of the gas at constant volume 
is readily found. Correction, of course, being made for change 
in volume of the containing vessel for change in temperature.* 
It must, of course, always be remembered that it is impossible 
to obtain absolutely accurate results by this method; since the 
thermal capacity of the gas is always small in comparison with 
that of the containing vessel. Still, Dr. Joly, who is the inventor 
of this method, has obtained fairly good results. 

67. Method of Clement and Desormes. In this method the 
gas, whose thermal capacity is sought, is contained in a large vessel 

* For a complete description, see "The Theory of Heat," by Preston, 
Chapter IV, Section V. 



ELASTICITIES AND THERMAL CAPACITIES OF GASES 71 

provided with a delicate manometer. When the contained gas 
has assumed the temperature of the surroundings, its pressure, 
which must differ from the atmospheric pressure, is carefully 
ascertained. When this has been done, a stop cock, having a 
large orifice, is opened and then closed after a very short interval 
of time. The time which elapses between the opening and closing 
of the stop cock must be so small that the change in the gas may 
be assumed adiabatic. During this change the temperature 
changes; i.e., there will be, either an elevation of temperature, 
if the pressure in the flask was initially less than the atmospheric 
pressure, or else, a diminution of temperature, if the pressure in 
the flask was initially greater than the atmospheric pressure. 
After the vessel and contents have again assumed the initial 
temperature, viz, the temperature of the surroundings, the pres- 
sure is again carefully noted. 

If, now, we denote by p\ the initial pressure of the gas, by v\ 
the corresponding volume per unit mass, by p the atmospheric 
pressure, which is also the pressure of the gas when the stop cock 
is open, and by v 2 the volume per unit mass after the Stop cock is 
closed, then since the change is assumed adiabatic, we may write 

pivi n =pv 2 n (1) 

Also, since the initial and final temperatures are the same, we have 

pm = p 2 v 2 ; (2) 

where p 2 is the pressure in the vessel after the temperature of the 
surroundings has again been assumed. 
From equation (1), we find 

W Pi 



from which 



logigM ...... (3) 

log {vi/v 2 ) 



72 % HEAT 

From equation (2) we find 

Vl = P2. 

V2 Vl 

from which 

log^ = log^ 2 . 

Substituting in equation (3) for log (wi/t^), its value, log (P2/P1), 
we obtain 

n Jog(p/pi) 
log (p 2 /pi) 

Or, expressing this in another form, we have 

logp-logpx ' . . (5) 

logp 2 -logpi 

Since p, pi, and P2 are known, n is determinate. In this 
manner, Roentgen found for dry air the value n = 1.405. 

This method is open to criticism, in so far that when the stop 
cock is opened, oscillations occur; and it does not necessarily 
follow that, at the instant of closing, the pressure in the vessel is 
equal to that obtaining outside. 

68. Isothermal and Adiabatic Elasticities. The ratio of the 
two thermal capacities of a gas is most accurately found by 
determining the speed of propagation of a disturbance through 
the gas. We will first show that the ratio of the two thermal 
capacities is numerically equal to the ratio of the two elasticities. 

From the statement of Boyle, we have, the temperature being 
maintained constant, 

pv = ki; (6) 

where h\ is a constant, depending upon the units chosen. By 
differentiation, we find immediately 



pdv-{-vdp = 0; 



ELASTICITIES AND THERMAL CAPACITIES OF GASES 73 
from which 



c 
the minus sign denoting merely that the volume decreases as the 

pressure increases. Now, the left-hand member of equation ( 

(7), is numerically equal to the ratio of change in unit stress to 

the corresponding change per unit volume; and is, therefore, 

by definition, the expression for the modulus of elasticity. Hence,/ 

for a gas obeying Boyle's law, the elasticity is numerically equal 

to the pressure. 

If now, we take the adiabatic equation, viz, 

Vv n = k 2 , (8) 

where k 2 is again a constant depending upon the units chosen, 
and differentiate, we find 



v n dp + nv n l pdv = ; 
vdp-\-npdv = 0; 



%?"* (9) 



from which 
and 



The left-hand member of equation (9) again expresses, accord- 
ing to definition, the modulus of elasticity. Hence, the modulus 
of elasticity when no heat is allowed to enter or escape, i.e., for 
adiabatic changes, is numerically equal to the product of the ratio 
of the two thermal capacities and the pressure. 

Denoting the isothermal elasticity by E t , and the adiabatic 
elasticity by En, we have 

En np 



ft V =n (I0) 



74 



HEAT 



From equation (10) we see that the ratio of the two principal 
elasticities is the same as the ratio of the two principal thermal 
capacities. 

69. Propagation of Wave Motion in an Elastic Medium. To 
properly appreciate how the ratio of the two elasticities, and hence 
the ratio of the two thermal capacities, of a gas is found from the 
speed of propagation of sound in the gas, it is essential to study 
the character of the motion by means of which sound is propagated 
in an elastic medium. 

Let AB, of Fig. 5, be a prism, of indefinite length and constant 
cross-sectional area, filled with a homogeneous elastic medium; 
and let the piston P have impressed upon it a constant acceler- 
ation toward the right. If the medium had absolutely no inertia, 
or were perfectly rigid, then the whole substance, between A and 

A abcdefgh 




yn 



Fig. 5. 



B, would suffer precisely the same displacement in a given interval 
of time. Due, however, to the inertia, the layer next to the piston 
will be compressed; the pressure of this layer now being greater 
than that of the medium in the undisturbed condition, it will 
react upon the second layer and compress this, which again in 
turn compresses the third layer, etc. Finally, when every layer 
throughout the prism, has been compressed by an amount such 
that its internal pressure is precisely equal to the applied pressure, 
then the acceleration of each layer will be the same and equal 
to that of the piston. 

Assume now, that the piston P is caused to vibrate periodically, 
with a small amplitude s. In tracing out a vibration we will 
begin by assuming the piston in the neutral position, moving 
toward the right, and the pressure of the medium, throughout, 



ELASTICITIES AND THERMAL CAPACITIES OF GASES 75 

the same as that in the undisturbed condition. As the piston 
moves toward the right, condensation takes place in the medium; 
the condensation being greatest for the layer in contact with the 
piston and becoming less as the distance from the piston increases. 
Suppose now, that when the piston has reached its maximum dis- 
placement s toward the right, the wave of condensation has reached 
the section represented by a; i.e., the pressure of the medium 
at the section a is the same as that in the undisturbed condition, 
and greater for all portions to the left of a. As the piston now 
begins to move toward the left, the wave of condensation continues 
moving toward the right; but, the pressure behind the piston 
begins to decrease, and by the time the piston has again reached 
its neutral position, the pressure of the medium directly in contact 
with the piston is the same as that in the undisturbed condition. 
The wave of condensation will, in the meantime, have traveled 
to the section b ; the distance ab being equal to Aa. The maximum 
condensation is now at a, and tapers off to zero from a to b and 
from a to A. As the piston now continues moving toward the 
left, the medium behind it becomes rarefied; and a wave of rare- 
faction travels toward the right. By the time the piston has 
reached its extreme left-hand position, the wave of rarefaction 
will have reached the section a, and the wave of condensation the 
section c; where the distances be and ab are equal. The maximum 
condensation now exists at b, and the maximum rarefaction at the 
piston. As the piston now begins its journey toward the right, 
the pressure behind it begins to rise, until it reaches the neutral 
position, when the pressure at the piston is equal to that of the 
medium in the undisturbed condition. In the meantime, the wave 
of condensation has traveled to the section d, where the distances 
cd and be are equal. Hence, during the time required by the piston 
to complete a period, the disturbance has traveled from A to d; 
and the conditions now existing are: Maximum condensation 
at c, maximum rarefaction at a, and at A, b, and d, the pressure 
is equal to that of the medium in the undisturbed condition. 



76 HEAT 

As the piston now continues moving toward the right, a wave 
of condensation moves toward the right from A, and also from 
d; and by the time the piston has completed its second cycle, 
the disturbance will have traveled to h; where the distances 
dh and Ad are equal. The condition of the medium between A 
and d is now at every section precisely the same as it was at the 
end of the first cycle. Also, the condition of the medium between 
d and h is precisely the same as it is between A and d; i.e., the 
condition at e is the same as at a, at / the same as at b, at g the 
same as at c, etc. At the end of the third cycle the disturbance 
will have traveled to the right of h, a distance equal to Ad = dh; 
and the condition of this portion will also be the same as the con- 
dition of the portions from A to d and from d to h. At the end 
of N cycles, the disturbance will have traveled through a distance 
equal to the product of N and Ad. 

The distance through which the disturbance travels while 
the piston goes through one cycle is called a wave length, and repre- 
sented by the letter X. Or, in other words, this is the distance a 
disturbance travels before conditions are beginning to be exactly 
reproduced. 

From the previous discussion, it is obvious that if there be 
performed N vibrations per unit time, and X is the wave length, 
then the speed of propagation is given by 

S = N\ (11) 

It is important to note that the distance through which the 
disturbance travels during a cycle depends upon the time consumed 
in performing that cycle; i.e., if the frequency — the number of 
vibrations per unit of time — be increased, then according to the 
discussion, the wave length will be proportionately less, such that 
the product of wave length and frequency is constant. This is 
fully verified by experiment for the speed of propagation of sound 
in gases. 



ELASTICITIES AND THERMAL CAPACITIES OF GASES 77 

70. Speed of Propagation in Terms of Elasticity and Density. 

Assume, as in Fig. 6, a cylinder of indefinite length and constant 
cross-sectional area A, filled with a homogeneous elastic medium 
whose density is p, and pressure per unit area in the undisturbed 
condition p. Assume further the frictionless piston P, having 
applied per unit area a pressure p+Ap; where Ap is a small frac- 
tional part of p. 

Now, as a matter of convenience, assume that the prism, repre- 
sented in Fig. 6, is divided into unit lengths, 1, 2, 3, etc., up to N; 
where N represents the distance the disturbance travels in a 
time t. The effect of the application of a pressure to the piston, 
in excess of the pressure of the medium, will be twofold; i.e., the 
medium will be compressed and also set in motion. It is evident 
that when any element of the medium in the prism has reached 



P+Ap 



N 



Fig. 6. 



a pressure per unit area equal to p-\- Ap, it cannot be further com- 
pressed, but will merely serve to transmit the applied stress to 
the next element. 

Let As be the amount of shortening a unit length undergoes 
while its pressure rises from p to p-\-Ap. The total shortening 
then, that the prism of length N undergoes in being compressed 
from the pressure p to a pressure p-\-Ap, and consequently the 
distance through which the piston moves during the time this 
change takes place, is 

d = NAs; 

and, since the time consumed to bring about this change is t, the 
speed with which the piston has been moving is 



d = NAs 
t t ' 



— - (12) 



78 HEAT 

But, at the instant that the pulse has passed through the distance 
N, all that portion of matter included in the length N, of the prism, 
is moving with a speed the same as that of the piston as given by 
equation (12); hence, its kinetic energy is 



W\ 



!+_NAp x (M,y (13) 



Since the total change in volume is AN As, and the average 
resisting pressure per unit area is p-\-Jp/2, the work, due to com- 
pression, is 

w 2 = (ANJs)(p+^ (14) 

But the total work done on the system must be equal to W1+W2; 
hence, 

(p+J P )ANJs=^pX (^) 2 + (v+ A i) ANAs, 

from which 

Ap 



(W-T 



(15) 



Now, N/t = S, the speed of propagation; and Ap/As, in the 
limit, represents the ratio of unit stress to unit strain, and hence, 
is equal to \l, the modulus of elasticity. Substituting, in equation 
(15), we have finally 



-^ 



s-V? <"» 

i.e., the speed of propagation of a disturbance through an elastic 
medium is numerically equal to the square root of the ratio of 
elasticity to density. 

The speed of propagation of sound in air is very readily deter- 
mined by experiment; and is found to be, at 0°C, very nearly 
332 meters per sec. The vibrations in a sound wave take place 



ELASTICITIES AND THERMAL CAPACITIES OF GASES 79 

so rapidly that the changes are sensibly adiabatic; hence [l, in 
equation (16), will be replaced by the adiabatic elasticity, and we 
have 



from which 



S = - ' np 



n = ^ (17) 

V 



Substituting numerical values, in equation (17), we find 

(33,200) 2 X 0.001293 , ._ , 
n= 1.0132X10 6 =L4 ° 6+ - 

The value 1.405 is generally used for dry air; but, for most 
practical purposes, 1.4 is sufficiently close. 



CHAPTER VII 
PROPAGATION OF HEAT 

71. Heat is transferred from one place to another in three 
distinct ways, viz, by radiation, by convection, and by conduction. 

72. Radiation. In Chapter II, we dealt with Newton's law 
of cooling, without considering in what manner the cooling takes 
place. As a matter of fact, in the cases considered, the cooling 
was due to two distinct phenomena. To illustrate this, we will 
consider a concrete case, viz, an incandescent lamp, which con- 
sists of a filament inside of a glass bulb; the bulb having been 
exhausted, so that the filament is practically in a vacuum. The 
propagation of heat from the filament to the glass bulb, that 
is, through a vacuum, is called radiation; or, in other words, 
radiation is the propagation of heat through space without the aid 
of any material substance. On the other hand, the dissipation of 
heat from the surface of the bulb is due, not only to radiation, 
but also convection; and the propagation of heat from the inner 
surface of the bulb to the outer surface is due to conduction. The 
propagation by convection and conduction will be considered 
later. 

73. Theory of Exchanges. Prevost, in 1792, promulgated 
the theory that there is a continual exchange of heat between 
bodies, even when they are at the same temperature. Prevost's 
theory may, perhaps, be best explained by means of the following 
illustration: Suppose a body suspended in a vessel, which has been 
completely exhausted. Assume further, that the walls of the enclo- 
sure are maintained at a constant temperature, and that the body, 
when first placed in the enclosure, has a temperature higher than 

80 



PKOPAGATION OF HEAT 81 

this. The temperature of the body will immediately begin to 
fall, due to its radiating heat to the walls of the enclosure; and 
this will continue until the temperature of the body is the same as 
that of the walls, when it becomes constant, and the body has 
apparently ceased radiating. If the temperature of the walls 
be now reduced, by immersing the vessel in a bath of lower tem- 
perature, the temperature of the body will fall and it will again 
be radiating heat. The body then, apparently, ceases to radiate 
heat when its temperature has fallen to that of the walls of the 
enclosure, and again begins to radiate heat when the walls are 
lowered in temperature ; and again ceases to radiate heat when its 
temperature has fallen to that of the walls, and so on indefinitely. 
If, initially, the temperature of the walls had been higher than 
that of the body, heat would have been radiated from the walls 
to the body. 

Now, according to the theory of exchanges, the body does not 
cease to radiate when its temperature has fallen to that of the 
walls; but, the body and the walls are continually radiating and 
absorbing heat. That is, it is assumed that, when the body is 
at a higher temperature than the walls, it is radiating heat more 
rapidly than it is absorbing heat, when at a lower temperature 
than the walls, it is gaining heat more rapidly by absorption than 
it is losing heat by radiation, and when at the same temperature 
the rates of radiating and absorbing heat are the same. 

Without being committed to this theory, it will be interesting 
to note certain conclusions which must necessarily follow from it. 

74. Emissivity. Experiments on radiant heat show that some 
bodies emit heat, other things being equal, more copiously than 
others. It is also found that bodies which are good radiators 
are also good absorbers. The capability which a body has for 
emitting heat is called its emissivity. 

Suppose now, that we have two bodies, placed in a space, imper- 
vious to heat. Then, according to Prevost's theory of exchanges, 
they will both radiate and absorb heat, even though they be at 



82 HEAT 

the same temperature. If now, one of the bodies absorbs heat 
more readily than it emits heat, its temperature will rise; this, 
however, is contradictory to experience. If, on the other hand, 
one of the bodies radiates heat more readily than it absorbs heat, 
its temperature will fall; which again contradicts experience. 
It therefore follows, if Prevost's theory of exchanges holds, that 
bodies have precisely the same capability for radiating heat that 
they have for absorbing heat. This appears to be in concordance 
with experiment. 

75. Stefan's Formula. We know that Newton's law of cooling 
is very limited in its application; i.e., it does not hold when the 
difference of temperature between the body under consideration 
and the surrounding medium exceeds 15°C. to 20 °C. In other 
words, it is only an approximate statement. Dulong and Petit 
performed a number of classical experiments by means of which 
they endeavored to determine the law of cooling. Their exper- 
iments were, however, limited in range of temperature; since, 
the maximum temperature reached was only about 240 °C. From 
their experiments, they deduced the equation, for the quantity 
of heat lost per unit time, 

Q = m¥(k d -1); (1) 

where Q is the quantity of heat, t the temperature of the enclosure, 
6 the difference in temperature between the enclosure and the 
radiating body, both measured on the centigrade scale, and m a 
constant, depending upon the substance and the nature of its 
surface. For k the value of 1.0077 was found. 

Stefan, from an examination of the results obtained by Dulong 
and Petit, deduced a formula for the loss of heat by radiation; i.e., 

Q = k(TS-T 2 4 ); (2) 

where k is constant, and T\ and I2 are the temperatures, as meas- 
ured on the ideal gas thermometer, respectively, of the radiating 
body and the enclosure. 



PKOPAGATION OF HEAT 83 

Equation (2) appears to give results, in accordance with exper- 
iments, up to temperatures of- about 1700°C. to 1800°C. It, 
however, appears from subsequent experiments, that Stefan's 
formula is not rigidly true; and consequently will require modi- 
fication. But, for practical purposes, Stefan's formula may be 
considered correct for the limits of temperature as stated. 

Considerable research work is still being done in regard to 
radiation at high temperatures; and whether, or not, a simple 
expression will finally be found which will be true for all temper- 
atures, is an open question. 

76. Convection. Referring again to the incandescent lamp, 
and considering the dissipation of heat from the surface of the 
bulb, we find that part of the heat is absorbed by the atmosphere 
surrounding the bulb, and the remainder is transferred by radia- 
tion. Due to the absorption of heat, the gases in contact with 
the bulb become heated and therefore change in density. This 
change in density destroys the equilibrium, in regard to pressure; 
and hence, currents are established, called convection currents, 
tending to restore equilibrium. In this manner, heat is conveyed 
from one portion of space to another by currents in the atmos- 
phere; i.e., the particles in contact with the bulb become heated 
and are replaced by particles at a lower temperature. These 
particles, in turn, become heated and are replaced by other 
particles; each particle carrying away a certain amount of heat. 

Since, in general, the density of liquids changes with change of 
temperature, it follows, that when a liquid is not of a uniform 
temperature throughout, convection currents will be established; 
and these will, of course, tend to bring about equilibrium. Thus, 
if a vessel containing a liquid, be heated at the bottom, the liquid 
in contact with the heated surface becomes less dense, rises, and 
is replaced by a portion of the liquid of higher density, which in 
turn becomes heated, is replaced by a denser portion, and so forth. 

77. Conduction. If a body, such as a metal rod, be heated at 
one end, then it is found that the temperature along the rod 



84 HEAT 

gradually rises; i.e., heat is transferred without the displacement 
of matter. Or, to put it in another way, heat is transferred from 
particle to particle, in a manner such that the particles maintain 
their relative positions. The propagation of heat through a solid 
is called conduction. 

Assume that we are dealing with a homogeneous body; bounded 
by two parallel plane surfaces, indefinite in extent, and that one 
surface is maintained at a temperature ti, and the other at some 
lower temperature T2. Then, after a certain time, steady condi- 
tions will be established. Consider now, the simplest case pos- 
sible, viz, a prism of constant cross-sectional area, normal to the 
two surfaces, and extending from one surface to the other. Now, 
since the two surfaces of the body are indefinite in area, we are 
justified in assuming that there is no lateral flow of heat; i.e., 
the heat flows through the prism in parallel stream lines, and 
the quantity of heat absorbed by the surface at a temperature 
ti, for a given interval of time, will be precisely 
equal in amount to the quantity of heat given 
off by the other surface, at a temperature T2, 
during the same interval of time. Or, in other 
words, the flow of heat through the prism will 
have become uniform; and the quantity of 
heat passing any section, parallel to the two 
surfaces, will be the same throughout. Then, 
as a fundamental principle, verified by experi- 
Fig. 7. ment, the temperature slope, or the rate of 

fall of temperature along the prism, is con- 
stant. Hence, if the distance between the two surfaces is 
represented by s, the temperature slope is 

r = ^; (3) 

where r is the temperature slope, or the rate of fall of temperature. 
The temperature at any point may be found as follows: Let, as 
in Fig, 7, a be the surface at a temperature of ti, b the surface at 



PROPAGATION OF HEAT 85 

a temperature of t2, and s the distance between the two surfaces. 
The rate of fall of temperature between the two surfaces is given 
by equation (3) ; and the fall of temperature from the surface a 
to the plane x, parallel to the two surfaces, is 

x ( v 

hence, the temperature of the plane x, is 

t* = ti— -(ti-t 2 ) (4) 

Theory indicates and experiment verifies that the quantity 
of heat which is transferred by a prism, such as has just been 
discussed, is proportional to the area of the exposed surfaces, to 
the time, and to the temperature slope. Stated symbolically 

QaoAtr; " . (5) 

where Q is the quantity of heat transferred, A the cross-sectional 
area of the prism, t the time, and r the temperature slope. To 
make statement (5) an equality, we must introduce a proportion- 
ality factor; i.e., 

Q = KAtr; 
from which 

*=! < 6 > 

K is the ratio of the quantity of heat, passing any section, to the 
product of the area of the section, the time, and the temperature 
slope at that section. This ratio is called the coefficient of con- 
ductivity of the substance; and, of course, differs for different 
substances. 

From equation (6) it follows that the coefficient of conductivity 
K, of a substance, is numerically equal to the quantity of heat which 
flows across a section of unit area, in unit time, when the tempera- 
ture slope is unity. In the c.g.s. system, and using the centi- 
grade scale, the coefficient of conductivity of a substance is numer- 



86 HEAT 

ically equal to the quantity of heat, measured in gram calories, 
which flows across a section 1 sq.cm. in area, in 1 second, when 
the temperature slope at the section is 1°C. per centimeter. 

78. Flow of Heat along a Bar. If a bar be maintained at a 
constant temperature at one end, and the remainder of the bar 
be exposed to a space of lower temperature, which is also main- 
tained constant, the fall of temperature along the bar will not be 
the same as that of the prism previously discussed. For, since 
the bar is at a higher temperature than the enclosure, it will con- 
tinually give up heat to the surroundings by radiation and con- 
vection. Eventually, heat will be supplied to every portion of 
the bar, by conduction, as rapidly as it is dissipated by radiation 
and convection. That is, the temperatures along the bar will 
finally assume steady values. But, as previously stated, the tem- 
perature slope along the bar will not be constant. For since, when 
a steady condition has been assumed by the bar, the quantity of 
heat which passes any section, for a given interval of time, is neces- 
sarily equal to the quantity of heat which is dissipated from the bar 
beyond that section, for the same interval of time, it follows that 
the quantity of heat which passes a section of the bar becomes 
less as the distance from the end, which is maintained at a con- 
stant temperature by the application of heat, increases. Hence 
since, other things being equal, the quantity of heat, which passes 
any section of the bar, is directly proportional to the temperature 
slope at that section, it follows that the temperature slope 
decreases with increase of distance from the heated end. 

79. Determination of Coefficient of Conductivity. Since, it 
is impossible to realize in practice those ideal conditions which 
were assumed in the discussion of the flow of heat between two 
parallel walls having areas of indefinite extent, recourse must 
be had to other methods. A bar maintained at a constant tem- 
perature at one end, and having the remainder exposed to a space 
of constant temperature, furnishes a convenient means for deter- 
mining the coefficient of conductivity. 



PROPAGATION OF HEAT 87 

To do this, we proceed as follows : After the bar has assumed 
a constant condition throughout, its temperature is ascertained 
at a number of definite points along it; this is most conveniently 
done by means of a thermo couple, which is calibrated by com- 
paring with a standard thermometer. The results are then plotted, 
differences of temperature between the bar and its enclosure as 
ordinates and distances along the bar as abscissas. The curve 
passed through the points so found, shows the difference of tem- 
perature between the bar and the enclosure, throughout the length 
of the bar; and the slope of the tangent, drawn to any point of 
this curve is numerically equal to the temperature slope at that 
section. This gives us r for equation (6) ; and A of this equation, 
viz, the area of the section, is determined directly from the dimen- 
sions of the bar. It now remains to determine Q/t, i.e., the quan- 
tity of heat which passes a section per unit time. To do this, a 
second experiment is necessary. The bar is now heated until 
its temperature is uniform throughout and slightly higher than the 
highest temperature on the curve for the rate of fall of temperature 
along the bar. The bar is then placed in the enclosure, under 
precisely the same conditions as obtained when the curve for the 
rate of fall of temperature was determined, and its temperature 
noted at definite intervals of time. From the data so obtained, 
a second curve is plotted, differences of temperature between 
the bar and the enclosure as ordinates and times as abscissas. 
The curve so obtained is the curve of cooling; and the slope of 
the tangent, drawn to any point of this curve, is numerically equal 
to the rate of change of temperature of the bar, with respect to 
time, for the particular difference of temperature between the bar 
and its enclosure at that time. 

Let it now be desired to determine the quantity of heat, which 
passes in a unit of time, some particular section of the bar, repre- 
sented by the point a, on the curve A, of Fig. 8. Curve A is the 
curve representing the temperatures along the bar, and curve 
B, the curve of cooling. If now, that part of the bar to the right 



88 



HEAT 



of a be divided into elements, such as ab, so short, that without 
appreciable error, the fall of temperature along the element may 
be considered constant, then the temperature of the element may 
be taken as the mean of the two temperatures at the points a 
and b. If this mean temperature be then projected across to the 
curve of cooling B, and at the point c, so found, a tangent be drawn, 
then the slope of this tangent is numerically equal to the rate 
of change of temperature with respect to time, for a difference 
of temperature equal in amount to the difference between that of 
the mean temperature of the element ab and its enclosure. If 




Fig. 8. 



now, we take the product of the thermal capacity of the element ab, 
and the rate of change of temperature just found, we obtain q/t, 
the quantity of heat lost, per unit of time, by the element ab at 
the instant when its temperature is defined by the point c. But, 
since the temperatures of the various parts of the element ab are 
constant, the mean temperature is a constant, and differs con- 
tinually from the temperature of the enclosure by an amount pre- 
cisely equal to the difference of temperature as found from the 
curve of cooling for the instant when the temperature is repre- 
sented by the point c. Therefore, the element ab is continuously 
losing heat, at a constant rate, equal in amount to the quantity just 
found from the curve of cooling for the temperature represented 



PROPAGATION OF HEAT 89 

by the point c. In a similar manner, the quantities of heat, escap- 
ing per unit of time, from the various elements to the right of the 
element ab, are found. Taking the sum of the quantities of heat 
so found, for all the elements to the right of ab, the quantity of 
heat Q/t, of equation (6), which passes the section a, in a unit of 
time, is found. From which, by substitution, K is found. 

Experiment shows, that, in general, the conductivity of solids 
decreases slightly with increase of temperature. 

80. Conductivity in Non-isotropic Substances. If there be 
a source of heat at a point in an isotropic substance, i.e., a 
substance having like physical properties in all directions, then 
other things being equal, heat will be propagated with equal 
speeds in all directions; and the temperatures at equal distances 
in all directions from the source of heat, at any instant, will be 
found the same. Or, in other words, the source of heat will be 
the center of spherical isothermal surfaces. On the other hand, 
substances which are non-isotropic do not conduct heat with equal 
speeds in all directions. As an example, the conductivity of Ice- 
land spar is greatest in the direction of the axis of symmetry, and 
equal in all] directions at right angles to this axis. It will be 
remembered that the coefficient of expansion for Iceland spar is 
also greatest in the direction of the axis of symmetry, and equal 
in all directions at right angles to this axis. 

81. Non-homogeneous Solids. Tyndall found, by experiment- 
ing with cubes of wood, that the speed of propagation of heat 
is greatest, in the direction of the fibers; i.e., parallel to the length 
of the tree, and least, parallel to the annual layers. And in a 
direction normal to both the fibers and the annual layers, i.e., 
radial to a section of a tree, a value was found for the conductivity 
slightly greater than that parallel to the annual layers; but, con- 
siderably less than that parallel to the fibers. Wood, however, 
on the whole is a very poor conductor in comparison with metals. 

It will be of interest here to note that the speed of propagation 
of sound through wood is different for the three directions; i.e., 



90 HEAT 

the speed of propagation is greatest, parallel to the fibers, 
least, parallel to the annual layers, and radial to a section of the 
tree, it is somewhat greater than it is parallel to the annual layers, 
but considerably less than that parallel to the fibers. 

82. Conductivity of Liquids. The determination of the co- 
efficient of conductivity of a liquid is attended by difficulties which 
are not experienced when dealing with solids. For, in the case of 
liquids, if we wish to determine the true conductivity, convection 
currents must be avoided. It is, therefore, necessary to heat the 
column of liquid from the top. It is impossible here, to consider 
all the necessary precautions which must be taken to insure 
accurate results. The principle involved, however, is precisely 
the same as for solids. That is, to determine accurately the 
temperature slope along the column and the quantity of heat 
passing a given section for a definite interval of time. 

83. Conductivity of Gases. The determination of the coef- 
ficient of conductivity of a gas is still more difficult than is the 
determination of the coefficient of conductivity of a liquid. For, 
in a case of a gas, not only must convection currents be eliminated, 
but radiation must also be taken into account. This makes it 
extremely difficult to obtain even fairly accurate results. 

As a matter of interest, the following coefficients of conductivity 
for a few substances are given. They are all expressed in the c.g.s. 
system with the gram calorie as the unit quantity of heat. That 
is, the numbers in the table represent in each case, the quantity 
of heat, in gram calories, which passes a section 1 sq.cm. in area, 
in 1 second, when the temperature slope is 1°C. per cm. 

Silver 1.01 Glass 0.002 

Copper 0.891 Firebrick 0.0017 

Aluminum 0.344 Cork 0.0007 

Zinc 0.265 Paraffine 0.0002 

Iron 0.167 Water 0.0014 

Mercury 0.0152 Ether 0.0003 

Ice 0.0057 Hydrogen 0.0004 

Granite 0.005 Air 0.000056 



PROPAGATION OF HEAT 91 

The student must always remember that the results given in 
the tables for the coefficients of expansion and conductivity must 
be taken as being only approximate. For, the physical properties 
of a substance depend very largely upon its chemical purity; and, 
furthermore, the properties any substance may manifest, will 
depend very largely upon its physical history and composition. 
This is especially true for alloys, such as brass, organic growths, 
such as cork, and complex compositions and mixtures, such 
as glass. 

It is interesting to note that, for metals, the order is the same 
for electrical conductivity as it is for thermal conductivity; i.e., 
good conductors of heat are also good conductors of electricity, 
and vice versa. However, there is not, as was at one time sup- 
posed, strict proportionality. 



THERMODYNAMICS 



CHAPTER VIII 
FUNDAMENTAL PRINCIPLES 

84. First Principle of Thermodynamics. The first principle 
of thermodynamics is merely the application of the principle of 
energy to the special case of mechanical work and heat; and may 
be stated as follows : When heat is converted into work, or work into 
heat, the ratio of the numbers representing the two quantities involved 
is a constant. The foregoing statement is, of course, the result 
of direct experiment. 

85. Second Principle of Thermodynamics. The second prin- 
ciple of thermodynamics is stated variously by different authors. 
Indeed, in some cases, the statement is preceded by discussions 
which involve almost the whole theory of heat. For our purposes, 
however, the statement first enunciated by Clausius will suffice. 
This statement is essentially as follows: Heat cannot pass from 
a body of lower temperature to one of higher temperature without 
the aid of some external agent. This statement, though not the 
result of direct experiment, is in conformity with our common 
experience. As an example, we know from experience that heat 
passes by conduction and radiation from regions of higher tem- 
perature to regions of lower temperature. To illustrate further, 
assume that we are dealing with two bodies A and B, and that the 
temperature of the former is lower than that of the latter; then 

93 



94 THERMODYNAMICS 

heat may be made to pass from A to B, by applying heat to A, 
until its temperature is the same as that of B, bringing the two 
bodies into contact, and by the further application of heat to A, 
heat will pass from it to B. Heat may also be made to pass from 
A to B, if work be first done on the former, such as compressing 
it, until its temperature is the same as that of B; then by bringing 
the two bodies in o contact and developing, by a further expendi- 
ture of work, more heat in A, heat will pass from it to B. But, 
until there is a tendency to raise the temperature of A above that 
of B, no heat will pass from the former to the latter. Assume, now, 
a third body, C, under compression and at the same temperature 
as A. By bringing the two bodies A and C into contact, and allow- 
ing C to expand against the external pressure, thus performing 
work, its temperature will fall and a certain quantity of heat will 
flow from A into C. The body C may now be removed from A, 
and compressed adiabatically until its temperature is equal to 
that of B } and then by bringing C into contact with B, and by a 
further expenditure of work on C, heat will flow from it to B. 
At the end of this process, C may be removed from B and allowed 
to expand adiabatically, and, if the various ranges have been 
properly chosen, it will at the end of this cycle of operations be 
in precisely the same condition as it was at the beginning. But, 
A now contains less heat, and B contains more heat than it did 
when the process began; and since C is in the same condition 
as it was at the beginning, heat has been transferred from a body 
of lower temperature to one of higher temperature by the aid 
of an external agent. 

86. Heat Motors. Heat Motors, or Heat Engines, are devices by 
means of which energy in the form of heat, is converted into 
energy, in the form of mechanical motion. 

All heat motors consist of three parts; viz, a source of heat, a 
working substance, and a refrigerator. Furthermore, all actual 
heat motors act periodically; i.e., operate on cycles; and for each 
cycle a certain quantity of heat is abstracted from the source by 



FUNDAMENTAL PRINCIPLES 



95 



the working substance, part of which is converted into work, 
and the remainder, neglecting radiation, etc., is rejected to the 
refrigerator. 

87. Simple Thermodynamic Engine. Before making a general 
demonstration it will be instructive to give, as an illustration, a 
simple concrete, though by no means economical, method for the 
conversion of heat into mechanical work. 

Assume that the source of heat is a reservoir of boiling water 
at 100°C, the refrigerator melting ice at 0°C, and the working 
substance, as depicted in Fig. 9, a metallic rod ab. One end of 




Fig. 9. 



the rod rests against the support SS f , and the other end against 
one of the teeth of the disk, supported on an axis through 0. A 
pawl P, pivoted on the support SS', also engages one of the teeth 
of the disk. Connected to this disk is a drum, having wound over 
it a cord, supporting a resistance R. Suppose now, that the length 
of the rod has been so chosen with respect to its coefficient of linear 
expansion, that when its temperature is raised from 0°C. to 100° 
C, it expands by an amount such that the disk turns through an 
angle equal to that subtended by a tooth. The resistance R, 
will suffer a certain displacement and the pawl will engage the 



96 THERMODYNAMICS 

next tooth. If the rod be now surrounded by a bath of melting 
ice, it will contract and engage the next tooth; the pawl, in the 
meantime, holding the disk in position. This process may be 
repeated indefinitely, until the resistance has been displaced 
through any desired distance. The cycle is then as follows: The 
rod at the temperature of melting ice is put into position, and sur- 
rounded by a jbath of boiling water at a temperature of 100 °C. 
In consequence of this elevation of temperature, the rod expands 
and turns the disk through a certain angle and in this manner does 
work in overcoming the resistance R. The heat taken from the 
source consists of two parts : One part being consumed in elevating 
the temperature of the rod, and is numerically equal to the prod- 
uct of the mass of the rod, its thermal capacity per unit mass, 
and the elevation of temperature. The other part consists of 
the heat equivalent of the work done in displacing the resistance 
R. The rod, now being disconnected and surrounded by melting 
ice, gives up to the refrigerator, in cooling from 100°C. to 0°C, 
an amount of heat precisely equal to that absorbed in being 
heated, without any external work being done, from 0°C. to 100° 
C. Therefore, the difference between the heat taken from the 
source and that given to the refrigerator is equivalent to the work 
done in displacing the resistance R. Since this completes a cycle 
it may be repeated indefinitely without any change in the relation 
of the quantities involved. 

If, now, we represent by Qi, the quantity of heat absorbed from 
the source, during a cycle, and by Q2, the heat rejected to the refrig- 
erator, then the external work done is 

W = J(Qi-Q 2 ); (1) 

where W is the external work done, and J the mechanical equiv- 
alent of heat. Since for every cycle the quantity of heat Qi 
has forever disappeared from the source, and only the part Q1 — Q2 
has been converted into work, it follows that, with the contrivance 



FUNDAMENTAL PRINCIPLES 97 

just described, it is impossible to convert all the heat, taken from 
the source, into external work. 

88. Heat of Expansion. As explained in Arts. 46 and 47, 
when a gas is heated and expands against an external pressure, 
the heat required is practically equal to that required to elevate 
the temperature of the gas, plus the heat equivalent of the exter- 
nal work done; i.e., the external work done, expressed in heat units, 
is practically equal to the heat of expansion. This, however, is 
by no means the case when a metal rod is heated and expands 
against an external pressure; for, in this case, the heat of expansion 
consists of two parts, viz, the heat equivalent of the external 
work done, and the heat required to expand the rod against its 
own inherent forces. The former may be called the external heat 
of expansion, and the latter the internal heat of expansion. 

In the present state of our knowledge we are unable to assign 
the proper relative values for the heat consumed in elevating the 
temperature of a substance and the internal heat of expansion; 
but, for most substances, the latter is a relatively large quantity. 

Since, now, in the cycle discussed in Art. 87, the internal heat 
of expansion is not recovered as work, but is rejected to the refriger- 
ator, it follows that such a contrivance cannot use heat eco- 
nomically. 

89. Carnot's Cycle. The first scientific discussion of a peri- 
odically acting thermodynamic engine is that due to Sadi Carnot, 
published in 1824. In this discussion, ideal conditions are assumed ; 
i.e., it is assumed that there are no losses due to radiation and 
friction. In other words, it was Carnot's object to show that 
under certain given conditions, assuming ideal processes, a definite 
fractional part of the heat taken from the source, by a periodically 
acting engine, is converted into work; and that, for the given 
conditions, this is the maximal amount of work that may be 
realized. The following demonstration will make this clear. 
Assume that we are dealing with any working substance whatso- 
ever, confined in such a manner that it may be put into contact 



98 



THERMODYNAMICS 



with either the source or the refrigerator, whose temperatures 
remain constant throughout the process, and that the conduction 
between the working substance and both the source and refrig- 
erator is perfect. Furthermore, for ideal conditions, it must be 
possible to insulate the working substance such that adiabatic 
processes may take place. 

Let now, in Fig. 10, the point A, on the p-v (pressure-volume) 
diagram, represent the pressure and volume of the working sub- 
stance at that part of its cycle when it is removed from the refrig- 
erator. Pressures are represented by ordinates and volumes by 




abscissas. The working substance is now insulated and com- 
pressed adiabatically, in consequence of which its temperature 
rises. This compression is continued until the temperature of 
the working substance is equal to that of the source, and its con- 
dition, as regards pressure and volume, is represented by the 
point B. The work done on the substance during this compres- 
sion is represented by the area under the curve; i.e., the area 
GABK. The working substance is now put into contact with 
the source and allowed to expand isothermally , by any desired 
amount, and its condition, as regards pressure and volume, at the 
end of this expansion, is represented by the point C. During this 



FUNDAMENTAL PRINCIPLES 99 

isothermal expansion a certain quantity of heat Qi, has been 
abstracted from the source, and work has been done by the working 
substance, represented by the area BCFK. The working sub- 
stance is now again insulated and allowed to expand adiabatically, 
in consequence of which its temperature falls. This expansion 
is continued until the temperature of the working substance has 
fallen to that of the refrigerator, and the work done by it, during 
this expansion, is represented by the area CDEF. The working 
substance is now put into contact with the refrigerator and com- 
pressed isothermally until its condition, as regards pressure and 
volume, is again represented by the point A. During this isother- 
mal compression, work was done on the working substance repre- 
sented by the area DEGA ; and, a quantity of heat Q2, was rejected 
to the refrigerator. 

Since now, the working substance, as regards pressure, volume, 
and temperature, is in precisely the same condition as it was at 
the beginning of the cycle, its intrinsic energy is also the same; 
it therefore follows, from the first principle of thermodynamics, 
that the difference between the heat abstracted from the source 
and that rejected to the refrigerator, expressed in mechanical 
units, is equal to the net work done. By an inspection of Fig. 
10 it is obvious that the net work done is represented by the area 
ABCD; and, from equation (1), we have 

Wi=J(Qi-Q 2 ); 

where Wi is the net work done. 

But the heat, expressed in mechanical units, abstracted from 
the source is 

W 2 =JQi; 

therefore, the ideal coefficient of conversion^ the maximal fractional 
part of the heat, abstracted from the source, which in an ideal 
process can be converted into work, is 

„ J(Qi-Q2) Q1-Q2 m 



100 THERMODYNAMICS 

The result, just found, has been deduced without making any 
assumption in regard to the nature of the working substance; 
it is therefore perfectly general. 

Suppose that the working substance suffers a physical change 
of state during the cycle; the foregoing demonstration still holds. 
For, since the working substance is in precisely the same condition 
as regards pressure, volume, and temperature at the end of the 
cycle as it was at the beginning, it follows that whatever physical 
changes of state have taken place during any part of the cycle, 
changes of a like kind must have taken place in the reverse order 
during some other part of the cycle; and hence, are balanced. 
Therefore, the difference between the heat taken from the source 
and that rejected to the refrigerator, expressed in mechanical 
units, is equal to the external work done. 

90. Since the relation, expressed in equation (2), was deduced 
without considering the properties of the working substance, it 
must be independent of those properties. There being, however, 
no other quantities involved in the right-hand member of this 
equation, excepting quantities of heat, and since these do not 
depend upon the properties of the working substance, they must 
be functions of the two temperatures. That is, the quantity of 
heat taken from the source must be some function of the tempera- 
ture of the source, and the quantity of heat rejected to the 
refrigerator must be some function of the temperature of the 
refrigerator. Just what values are to be assigned to these func- 
tions must be determined for some specific case, which is con- 
sistent with the demonstration. 

91. Carnot's Cycle a Reversible Process. The ideal cycle just 
described is a reversible process. For, if the working substance, 
at that part of its cycle when its condition, as regards pres- 
sure and volume, is represented by the point A, Fig. 10, and its 
temperature is the same as that of the refrigerator, is put into 
contact with the refrigerator and allowed to expand isothermally 
to the point D, it will abstract from the refrigerator, a quantity 



FUNDAMENTAL PEINCIPLES 101 

of heat Q2, and do an amount of external work, represented by the 
area ADEG. The working substance is then insulated and com- 
pressed adiabatically , in consequence of which its temperature will 
rise; let this be continued until its temperature is the same as 
that of the source, and its condition, as regards pressure and vol- 
ume, is represented by the point C, and an amount of work, 
represented by the area FEDC, has been done on the working 
substance. The working substance is now put into contact with 
the source, and compressed isothermally until its condition, as 
regards pressure and volume, is represented by the point B, 
a quantity of heat Qi being rejected to the source, and an amount 
of work, represented by the area FCBK, has been done on the work- 
ing substance. The working substance is now insulated and 
allowed to expand adiabatically until its temperature has fallen 
to that of the refrigerator; its pressure and volume being the same 
as at the beginning, and the external work, represented by the area 
BAGK, having been done by it. Taking the sum, we find that the 
work done by the working substance is represented by the area 
KBADE; and the work done on the working substance is repre- 
sented by the area BKEDC. Finally, the net work done on the 
working substance is represented by the area ADCB. But, 
during this process, the quantity of heat Q2 has been taken from 
the refrigerator, and the quantity of heat Qi has been transferred 
to the source. 

Since now, the working substance is in precisely the same 
condition as regards temperature, pressure, and volume, as it 
was initially, it follows that the difference between the heat 
rejected to the source and that taken from the refrigerator, 
expressed in mechanical units, is equal to the net work done on 
the working substance ; i.e., 

W = J(Qi-Q 2 ). 

92. It will now be shown that, for a given source and refrig- 
erator, an engine operating on the Carnot cycle, that is, a reversible 



M)2 THERMODYNAMICS 

engine, converts into work as large a fractional part of the heat 
taken from the source as is possible under the assumed conditions. 
To do this, we assume that we have two engines A and B, operating 
between the same source and refrigerator, the former acting 
direct and driving the latter, which is running reversed. Let 
Ha and H a " be, respectively, the heat taken from the source and 
that rejected to the refrigerator by the engine A during a given 
interval of time; and likewise, let H b ' and H b " be, respectively, 
the heat transferred to the source and that abstracted from the 
refrigerator by the engine B, during the same interval of time. 
Assume, now, that the engine A, which is non-reversible, can convert 
a larger fractional part of the heat taken from the source into 
work than could the engine B if it were running direct. We then 
have 

TJ ' rr // tj t tj // 

tig —tig tit —lib , Q s 

H a ' > H»' {6) 

Also, the work done by the engine A must be equal to the work 
done on the engine B, since the former is driving the latter; hence, 
we have 

W = J(Hg'-H a ")=J(Hb'-H b ") (4) 

From equation (4) it follows that the numerators of the inequality, 
expressed by statement (3), are equal; hence 

Hg <Hb . 

Also, from equation (4), we find 

TJ ' TJ ' — TJ 1 1 TJ ff . 
lib —fla —Jib —tl a , 

hence, since Hb is greater than Ha, Hb' must be greater than 
Hg' . But, with a reversible engine, operating on the Carnot 
cycle between a certain source and refrigerator, the quantities 
of heat involved are always the same for a definite amount of 
work, no matter whether the engine is acting direct or reversed; 



FUNDAMENTAL PRINCIPLES 103 

it therefore follows that under the assumed conditions, the source 
must be gaining heat, since the quantity of heat H a ' taken from 
it by the non-reversible engine is less than the quantity of heat 
H b f rejected to it by the reversible engine. Likewise, since the 
quantity of heat H a " rejected to the refrigerator, by the non- 
reversible engine, is less than Hb" } that taken from it by the revers- 
ible engine, it follows that the refrigerator is continually losing 
heat. Hence, under the assumed conditions, we have a system 
in which heat is being transferred from a body of lower temper- 
ature to one of higher temperature, without the aid of an agent 
external to the system. Since this, however, contradicts the 
second principle of thermodynamics, we must conclude that the 
assumption made, viz, that any engine can convert into work 
a larger fractional part of the heat taken from the source than is 
possible by means of a reversible engine, operating on a Carnot 
cycle between the same source and refrigerator, is in error. There- 
fore, a reversible engine converts into work as large a fractional 
part of the heat taken from the source as is possible under the 
given conditions. 

We may, however, consider this in another manner. Assume 
that there is a third engine, operating between the same source 
and refrigerator, abstracting heat from the source, and rejecting 
heat to the refrigerator at a rate such that both the source and 
refrigerator are maintained at a constant temperature. This 
third engine may then be employed in doing external work, and 
we have a system which is doing work without the expenditure 
of energy. This, however, contradicts the principle of energy; 
hence we must again conclude that the original assumption is in 
error. Hence the conclusion that, for any given conditions, no engine 
can convert into work a larger fractional part of the heat taken from 
the source than that converted into work by a reversible engine. 

93. Reversible Engine as a Standard. It must be remembered 
that all processes so far discussed, in this chapter, are ideal proc- 
esses and cannot be realized in practice ; in other words, since the 



104 THEEMODYNAMICS 

cycles of an actual thermodynamic engine, are attended by friction 
and radiation, they are necessarily irreversible. As a matter of 
fact, as stated in Art. 27, all processes are irreversible. Reversible 
processes are merely ideal; i.e., conceptions of perfect operations. 

When we speak of the efficiency of a mechanical contrivance, 
as being the fraction p, we simply mean that the output is the 
fractional part p of the input. And the closer p approaches unity, 
the nearer the machine is considered to be to perfection. But 
here again, our standard is an ideal one; i.e., we are comparing 
our actual machine with one that is ideally perfect. Since it 
has been shown that a reversible engine, operating on the Carnot 
cycle, between a given source and refrigerator, converts into work 
as large a fractional part of heat taken from the source as possibly 
can be converted into work under the given conditions, we are 
justified in taking this engine as a standard with which to compare 
the performance of actual engines. 

94. Carnot* s Cycle with a Perfect Gas as a Working Sub- 
stance. Assume that we have confined in a cylinder, by means 
of a frictionless piston, a perfect gas, and that there is a source 
of heat at a temperature T\, and a refrigerator at a temperature 
T r 2. Let the condition of the gas, as regards pressure and volume, 
be represented by the point A, Fig. 11, when put into contact 
with the source; pressures being represented by ordinates and 
volumes by abscissas. We will, furthermore, assume ideal 
conditions; i.e., perfect conduction for the isothermal processes, 
perfect insulation so that adiabatic processes may take place, and 
no losses. Consider now, the four processes as follows : 

(1) The cylinder containing the gas at the pressure pi, volume 
vi, and temperature Ti, is put into contact with the source of 
heat at temperature T\, and the gas is allowed to expand isother- 
mally by any desired amount, say to the point B; its pressure 
now being p2 and volume V2. During this expansion work is 
done on the piston by the gas, measured by the area under the curve 
AB, and a quantity of heat Qi is taken from the source. It being 



FUNDAMENTAL PRINCIPLES 



105 



assumed that the temperature of the source during the abstraction 
of the heat Qi, remains constant; this may be brought about by 
supplying heat to it at a proper rate. 

(2) The cylinder is now removed from the source, is perfectly 
insulated, and the gas is allowed to expand adiabatically , in con- 
sequence of which its temperature falls, due to the fact that the 
external work is done at the expense of the intrinsic energy of 
the gas. This expansion is continued until the temperature of 
the gas has fallen to I2, that of the refrigerator; in the meantime, 



?<p.,V 




(P«,v 4 ) 



(p s , v *> 



V 

Fig. 11. 



work has been done on the piston by the gas, measured by the area 
under the curve BC. The pressure is now 7^3 and the volume v%. 

(3) The cylinder is now put into contact with the refrigerator 
and the gas is compressed isothermally , until its pressure is p4, 
and volume v±, as represented by the point D. During this com- 
pression a quantity of heat Q2 is developed, and work is done by 
the piston on the gas, measured by the area under the curve DC. 
It being assumed that the temperature of the refrigerator, during 
the absorption of the heat Q2, remains constant; this may be 
brought about by abstracting heat from it at the proper rate. 

(4) The cylinder is now removed from the refrigerator, is 
perfectly insulated, and the gas is compressed adiabatically until 



106 THERMODYNAMICS 

its temperature is T\, that of the source, and its pressure and vol- 
ume are, respectively, pi and vi. During this compression work 
was done, by the piston on the gas, measured by the area under the 
curve AD. The gas being now in precisely the same condition 
as it was initially, its intrinsic energy must also be the same. 

Since, now, A and B are on the same isotherm, and likewise, 
C and D are on the same isotherm, we have, from the character- 
istic equation, 

piVi = p 2 V2 = RT 1} (5) 

and 

P3V3 = P4V4 = RT2 (6) 

Also, since B and C are on the same adiabatic, and likewise, 
A and D are on the same adiabatic, we have 

p 2 V2 n = P3V 3 n , (7) 

and 

PlVi n =P±V4: n (8) 



From equation (5) we find 



and 



From equation (6) we find 



RTi , . 



RTi /inN 

p2= ^- (10) 



v *=~w (11) 

and 

p* = -^~ w 

Substituting the value of P2 as given in equation (10), and that 
of pz as given in equation (11), in equation (7) we find 

V2 V3 



FUNDAMENTAL PEINCIPLES 107 

from which 

1 

V3 /TAn-l 



n \T 2/ < 13 > 

Again, substituting in equation (8) the values of pi and p4,as given 
by equations (9) and (12), we find 

RT 1 n RT 2 n 

Vl n = V4 n . 

Vl V± 

from which 

vr\T 2 ) w 

From equations (13) and (14) it follows that 



Vz 


_V4 


V 2 


Vl 


V3 = 


_V2 


V4 


Vl 



from which 

(15) 

Equation (15) shows that the volume at C must be to the 
volume at D, as the volume at B is to the volume at A, so that 
when the gas is compressed adiabatically from D, it will come to 
the point A. 

Since we are dealing with a perfect gas, its intrinsic energy is 
a function of the temperature only, and is, therefore, independent 
of pressure and volume. Therefore, the work done by the gas in 
going along the adiabatic from B to C, is exactly equal to the 
work done on the gas in going along the adiabatic from D to A. 
Hence to obtain the net work done during the cycle, and the quan- 
tities of heat involved, it is only necessary to consider the two 
isothermal processes; viz, the heat abstracted from the source, 
and the external work done, by the gas in going from A to B, 
along the isotherm Ti, and the heat rejected to the refrigerator, 



108 THERMODYNAMICS 

and work done, on the gas, in going along the isotherm T 2 from 
CtoD. 

Since the temperature, during the isothermal expansion, from 
A to B is constant, it follows that the heat abstracted from the 
source is directly proportional to the external work done. Hence 
we have 



Qi=A I pdv: 

J n 



where A is the heat equivalent of a unit of work. But p, for 
any part of this process is equal to RTi/v; hence 

"2 dv 



r v2 d\ 

Q 1 =ART 1 - 
Jh v 



=ART 1 \og V ^ (16) 

By similar reasoning we find, that the heat rejected to the 
refrigerator, during the isothermal compression, in going from 

C to D, is 

fn 

Q 2 = A pdv. 

But, for any part of this process p is equal to RT2/V; hence 

* V3 dv 



Q 2 = ART 2 



r v3 dv 

Jv 4 V 



=ART 2 \og V ^- (17) 

V4 

The net work done, measured in heat units is, by condition, 
proportional to the difference between the heat taken from the 
source and that rejected to the refrigerator; hence 

AW = Qi-Q 2 = AR(T 1 log V ^-T 2 log^); 

\ V\ V4J 



FUNDAMENTAL PRINCIPLES 109 

and the ideal coefficient of conversion, since Qi has forever dis- 
appeared from the source, is 



Qi-Qj 



W^log^-^log^) 
\ Vi ° v 4 / 



Ql AR Tl log** 

Vi 

from which, since by equation (15), 02/01 = 03/04, we find 

Qi-Q2 _ T 1 -T 2 . 

Y1 ~ Qi -Ti (18) 

Equation (18) shows that, for a perfect gas operating on a 
Carnot cycle, the ideal coefficient of conversion is the ratio of the 
difference in temperature of source and refrigerator, to the tem- 
perature of the source, as measured on the ideal gas thermometer. 

Equation (18) is usually written in the following form: 

1=-^; • (19) 



where S is the temperature of the source and R that of the refrig- 
erator, both being measured by means of the ideal gas thermometer. 
95. A little consideration will show that the foregoing discussion, 
and result obtained, is perfectly consistent in every way with that 
of the Carnot cycle using any working substance. We are there- 
fore justified (Art. 90) in assuming that even under ideal conditions 
the maximum quantity of work that can be realized from an 
engine working between a given source and refrigerator and ab- 
sorbing the quantity of heat H from the source, is 

W=Jh(^) (20) 



Writing equation (19) in another form, we have 

1 R 



110 THERMODYNAMICS 

from which it is obvious that, for y] to approach unity, R must 
either approach zero, or S must approach infinity. Experience, 
however, shows that it is not economical to attempt to maintain 
the refrigerator at a temperature lower than that of the surround- 
ings. Also, as the temperature of the source is increased, a point 
is soon reached for which radiation and pressures become excess- 
ive, and lubrication becomes difficult. It therefore follows, 
with conditions such as obtain on the earth's surface, that even 
a perfect engine can convert only a small fractional part of the 
heat, taken from a source, into work. 

96. Reversible Engine and Refrigeration. An engine operating 
in a reverse order, i.e., one that is taking heat from a body of 
lower temperature, transferring heat to a body of higher temper- 
ature, and absorbing external work, constitutes a refrigerating 
machine. Let, for any given time, Hi be the quantity of heat 
transferred to a body of higher temperature, and H2 the quantity 
of heat abstracted from a body of lower temperature, by a per- 
fectly reversible engine; i.e., a perfect refrigerating machine. 
We will then have the following relation : 

H1—H2 _ S—R . . 

#1 " s {2l) 

From equation (20) we have, for the amount of work that must 
be done, to transfer the quantity of heat Hi, to the body of higher 
temperature, 

W = JHi^^ (22) 

In general, however, in the case of refrigerating machines, we are 
concerned principally with the work that must be done to bring 
about a certain absorption from the body of lower temperature; 
i.e., the amount of refrigeration. It is therefore advisable to 
deduce an expression for the amount of work that must be done 
in terms of H 2 , the quantity of heat taken from the body of lower 



FUNDAMENTAL PRINCIPLES 111 

temperature, instead of the quantity of heat Hi, rejected to the 
body of higher temperature. 
From equation (21) we find 

H2 = R 

Hx S' 
from which 

#i=# 2 § (23) 

Substituting in equation (22) the value of Hi, as given by equation 
(23), we find 

W = JH 2 ^j^; ...... (24) 

which gives the desired relation. 

97. In Art. 95, it was stated that it is not economical to attempt 
to maintain the temperature of the refrigerator lower than that 
of the surrounding medium. We are now prepared to demon- 
strate this mathematically. 

Let Hi be the heat taken from the source, at a temperature 
S, and let Ri be the temperature of the surroundings. If then the 
temperature of the refrigerator be also Ri, the work that, under 
perfect conditions, may be realized is 

Wi=JHi^f±. ..... (25) 

Assume now, that the refrigerator, by means of a reversible engine, 
is maintained at some temperature R2, lower than Ri. The work 
that can now be realized, by means of a perfect engine, is % 

W 2 = JhJ^^ (26) 

Subtracting equation (25) from equation (26), member by mem- 
ber, we obtain, due to lowering the temperature of the refriger- 
ator, for the gain in work, 

W 2 -W 1 =J^(R 1 -R 2 ) (27) 



112 THERMODYNAMICS 

To maintain the temperature R 2 we must, by means of a revers- 
ible engine, abstract heat from the refrigerator at the same rate 
that the direct engine is rejecting heat to it, and transfer heat to 
the surroundings. The heat rejected by the direct engine is 

H 2 = H 1 -AW 2 =H 1 -H 1 ^~^ = H 1 ^. . . (28) 

The work that must be expended in transferring this quantity 
of heat from the body of temperature R 2 , to the surroundings at 
a temperature Ri, is 

= J I ^(R 1 -R 2 ) (29) 

By comparing equations (29) and (27), it is obvious that, 
even under ideal conditions, the amount of work that must be 
done by the reversible engine, to maintain the temperature of the 
refrigerator, below that of the surroundings, is equal to the gain 
in work by the direct engine, due to the lower temperature of 
the refrigerator. It therefore follows that, even without consid- 
ering losses, there can be nothing gained by attempting to have the 
temperature of the refrigerator lower than that of the earth's sur- 
face. As a matter of fact, if the temperature of the refrigerator is 
lower than that of the surroundings, heat will continually pass 
from the surroundings to the refrigerator, and the reversible 
engine must do an amount of work greater than that given by 
equation (29). Furthermore, due to imperfections of the engines, 
the gain in work realized by the direct engine will be less than that 
specified by equation (27), and the work that must be done on the 
reversible engine will be greater than that specified by equation 
(29) ; hence, there is a decided loss when the refrigerator is main- 
tained at a temperature lower than that of the surrounding media. 



FUNDAMENTAL PRINCIPLES 



113 



98. Thermodynamic Scale of Temperatures. The thermo- 
dynamic scale of temperatures, which was first proposed by Lord 
Kelvin, will be made clear by the following considerations. Assume 
a series of n perfect heat engines arranged in such a manner that 
the refrigerator of the first engine is the source of the second 
engine, the refrigerator of the second engine is the source of the 
third engine, etc., and furthermore, that the heat rejected by any 
engine is absorbed by the engine next lower in the scale. To show 
that, if the difference in temperature between source and refrig- 




erator for the various engines is the same, they are all doing the 
same amount of work. 

Let, as in Fig. 12, the two adiabatics, AB and CD, be cut by 
the isotherms T 1} T 2 , T 3 , etc., such that the temperature intervals 
are all equal, and each equal to t; i.e., 

T 1 -T 2 = T2-Tz = T n -T n + 1 = 'z. 

The ideal coefficients of conversion for the various engines, begin- 
ning with the first, then are 



TV zy 5y 



T 



(30) 



114 THERMODYNAMICS 

If H is the quantity of heat absorbed by the first engine from its 
source, during a given interval of time, then 

b-b—tT - h T! 

is the heat rejected to its refrigerator, and absorbed by the second 
engine, during the same interval of time. In a similar manner, 
the quantity of heat supplied to the third engine is 

rp rp rp rp rp 

jjl2 rri2 v l2— i3 = TTJ3 
" m ■" rp S^ rp •" rp • 

The quantities of heat supplied to the various engines, beginning 
with the first, then are 

Uy U Y'y U T~> ' ' ' n ~f — » n jT' • • \ 6i -J 

Since, now, the work done by any engine of the series is equal 
to the product of its ideal coefficient of conversion and quantity 
of heat, expressed in mechanical units, absorbed by it, it follows 
from expressions (30) and (31), that all the engines are doing the 
same amount of work; i.e., 

is the work done by each engine of the series. 

The results just deduced, being independent of the properties 
of any substance, a thermodynamic scale of temperature may be 
established in the following manner: Assume a series of n heat 
engines, working between a given source of temperature T\, and 
refrigerator of temperature T n + i, in such a manner that the n 
engines are all doing the same amount of work, and each engine 
is absorbing the heat rejected by the engine next higher on the 
scale. If we then designate the difference of temperature between 



FUNDAMENTAL PRINCIPLES 115 

the source and refrigerator of any one of these ideal engines, as 
a unit of temperature, we will have a scale of temperatures inde- 
pendent of any substance, and depending only upon the perform- 
ance of a perfect engine. But, from the discussion just given, 
we found that by assuming the temperature intervals, as measured 
on the ideal gas thermometer, equal, the engines were all doing 
the same amount of work; hence, the thermodynamic scale is 
identical with that of an ideal gas thermometer; and differs but 
slightly, for temperatures not exceeding 500 °C, from those as 
found by means of the ordinary gas thermometer. 



CHAPTER IX 

STEAM AND STEAM ENGINES 

99. The proper design of a heat engine presupposes, on the 
part of the designer, a knowledge of the construction of mechan- 
ical contrivances; i.e., how to construct a machine which shall 
withstand the stresses imposed upon it in the performance of 
its duties, with the lowest cost. The expression, lowest cost, must 
not be interpreted as meaning lowest first cost; but it must be 
understood to mean that the interest^on the capital invested, 
for both machinery and ground rent, plus depreciation, plus 
cost of power lost, must be a minimum. This part of the subject 
comes under the heading of machine design; and, properly speak- 
ing, has nothing to do, except in so far as fuel economy is affected 
by the design, with the subject of thermodynamics. But, a thorough 
knowledge of the characteristics of the working substance and the 
changes it undergoes, during its various stages, is fully as impor- 
tant, if not more so, in the designing of an engine, as is a knowledge 
of machine design. It is for this reason, since steam is so widely 
used as a working substance, that so much research work has been 
done, to accurately determine its characteristics. 

100. Steam Operating on Carnot's Cycle. Assume that we 
are dealing with a unit mass of water, at a temperature I2, which 
corresponds to that of the refrigerator, and let its condition, 
as regards pressure and volume, be represented by the point D 
of Fig. 13. The water is compressed adiabatically until its tem- 
perature is T\, that of the source, and its condition, as regards 
pressure and volume, is represented by the point A. If the water 

116 



STEAM AND STEAM ENGINES 



117 



is now placed into contact with the source, and the pressure is 
maintained constant, vaporization will take place. Assume this 
to be continued until all the water has been converted into satu- 
rated steam, whose condition, as regards pressure and volume, 
is represented by the point B. The steam is now allowed to expand 
adiabatically until its temperature has fallen to T2, that of the 
refrigerator; its pressure and volume being now represented by 
the point C. During this adiabatic expansion a certain amount 
of condensation, which will be discussed later, has taken place. 
The mixture of steam and water is now put into contact with the 



\ 


.Tt 


B 
To 




D L 















V 
Fig. 13. 



refrigerator and compressed isothermally until complete conden- 
sation has taken place, and its condition, as regards pressure and 
volume, is again represented by the point D. Since, now, the 
condition of the working substance, as regards temperature, 
pressure, and volume, is precisely the same as it was initially, 
its intrinsic energy is also the same. Therefore, the net work 
done during the cycle is measured by the area DABC. Further- 
more, since the process is ideally reversible, the ideal coefficient 
of conversion is 

T l -T 2 



f\ = 



Tx 



the same as previously deduced for any working substance. 



118 



THERMODYNAMICS 



101. Relation of Temperature and Density of Saturated Steam. 

It is frequently of prime importance to know the density of satu- 
rated steam for a given temperature; and it being difficult to 
determine this relation by direct experiment, it will be shown how 
it is found from the relation of pressure and temperature of a 
saturated vapor, this being easily determined by direct experiment. 
To show how to determine the relation of temperature and density 
of the saturated vapor of a substance, it will be assumed that we 
are dealing with a unit mass operating on a Carnot cycle, as just 
described, and an indefinitely small difference of temperature, 
AT, between source and refrigerator. This is represented dia- 
grammatically in Fig. 14, where T-\-AT is the temperature of the 



T+ AT 



^ 



V 
Fig. 14. 



source, and T the temperature of the refrigerator. This being 
a reversible process, the ideal coefficient of conversion is 



Y) = 



AT 



T+AT> 



which, in the limit, becomes 



dT 



T) = 



If the quantity of heat taken from the source, in going from A 
to B, is Q, then the work done is 



W = JQ 



dT 



(1) 



STEAM AND STEAM ENGINES 



119 



250 



225 



200 



175 



• 150 



5l25 

Q. 



100 



75 



50 



25 



■ 










































































































































































































/ 




































































/ 




































































/ 










RELATION 






























OF 


















1 










TEMPERATURE AND PRESSURE 


















/ 






























/ 






























/ 












SATURATED STEAM 


















' 




























/ 




































































/ 




































































/ 




































































1 


































































I 












































































































































































































/ 


































































I 




































































/ 




































































/ 






































































































































1 




































































/ 




































































/ 


































































1 


' 






































































































































1 


































































) 








































































































































1 


































































j 






































































































































/ 




































































/ 




































































/ 


































































/ 




































































/ 












































































































































































































































































































































/ 




























- 






































/ 
































































/ 
































































^ 


/ 

















































































































































































































































100 



200 300 

TEMP. IN DEGREES FAHR. 



400 



1 20 THERMODYNAMICS 

The work done during the cycle may also be expressed in terms 
of the initial and final volumes and the change in pressure dp, 
corresponding to the change in temperature dT. That is, if a 
is the volume of unit mass of the liquid, and s the volume of unit 
mass of saturated vapor, then the work done, during the cycle, 
is 

W = (s-a)dp (2) 

Now the right-hand members of equations (1) and (2) must be 
equal; since they are expressions for the same amount of work, 
hence 

dT 

JQ-y = (s-°)dp; 



from which 



and 



JQ K dT 



•«+f *§ « 

The quantity Q, in equation (3), represents the quantity of 
heat required to convert unit mass of the liquid into a saturated 
vapor at the temperature T, and may be replaced by r, the heat of 
vaporization; hence, equation (3) becomes 

S = °+T X Jp (4) 

In equation (4), a, the volume of unit mass of the liquid, for 
the temperature T, is readily found by experiment; and likewise 
r, the heat of vaporization. dT I dp is found from the curve giving 
the relation of temperature and pressure of the saturated vapor. 
Hence, since J, the mechanical equivalent of heat, is known, 
s is determinate ; and the reciprocal of this gives the density of 
the saturated vapor. 



STEAM AND STEAM ENGINES 121 

As a matter of interest, the curve showing the relation of 
temperature and pressure, for saturated steam, is given on 
page 119. 

102. Perfect Steam Engine and Boiler. In the previous dis- 
cussions it has been assumed that all of the heat is taken in at 
the highest temperature. This, however, is by no means the case, 
even under perfect conditions, with a steam engine and boiler. 

For the present, we will confine ourselves to the operation 
of a reciprocating engine, which has supplied to it saturated steam 
from a boiler. The reciprocating engine consists essentially of 
the following parts: A source of heat, the boiler, where steam is 
generated under a constant pressure, and hence, at a constant 
temperature, a cylinder and piston, and a refrigerator, or condenser, 
at constant temperature, by means of which the steam, after 
expanding and doing work against the piston, is converted into 
water and returned to the boiler. The cycle of operations is 
as follows: The piston P is at the position as represented in the 
diagram, Fig-. 15, and the condition of the steam, as regards pres- 
sure and volume, is represented by the point A, the point of 
admission. That is, at this point, the valve in the pipe connect- 
ing the boiler with the cylinder is opened, and steam is freely 
admitted. The piston advances to the point B, while vapori- 
zation takes place at the temperature T\. To simplify matters, 
we will assume that we are dealing with unit mass of water and 
that complete evaporation has taken place when the volume is 
represented by the point B. The quantity of heat then, taken 
from the boiler, is n, the heat of vaporization at the temperature 
T\. The line of admission, A B, is a straight line and parallel to 
the axis of volumes, since vaporization has taken place at constant 
temperature; and hence, at constant pressure. The external 
work done, during this advance of the piston, is measured by the 
area ABFE. The point B is the point of cut-off; i.e., the admission- 
valve is closed, and the steam is allowed to expand adiabatically 
until its temperature has fallen to T 2 , that of the condenser. 



122 



THERMODYNAMICS 



In the meantime, the external work, measured by the area BCGF } 
has been done. The exhaust-valve now opens, and the steam 
remaining in the cylinder, is compressed isothermally, in contact 
with the condenser, until complete condensation has taken place, 
and the work represented by the area DCGE, has been done by 
the piston. The condensed steam, at the temperature T2, is 
returned to the boiler and heated from the temperature I2 to 
that of T\, thus completing the cycle. The net work done during 



A T i B 




d !k t 2 ^^ ^ 


E If _[g 


v i 
1 




2 




^ i 





Fig. 15. 



this cycle is evidently measured by the area ABCD, and is neces- 
sarily less, as will now be shown, due to not taking in all the heat at 
the maximum temperature, than that which could be realized by 
a Carnot cycle. 

The maximum amount of work that could be realized from 
an engine taking in an elementary quantity of heat dQ, at the 
temperature T, between the temperatures T2 and T\, operating 
on a Carnot cycle, and rejecting heat to the condenser at the tem- 
perature T2, is 

T-T 2 



dW = JdQ 



(5) 



STEAM AND STEAM ENGINES 123 

But, dQ is equal to cdT; where c is the thermal capacity and dT 
the change in temperature. And, since we are dealing with unit 
mass of water, dQ is practically equal to dT; since for water c 
is almost constant and equal to unity. Therefore, equation (5) 
may be written 

dW=J^^dT; ....... (6) 



from which we obtain, for the total work that could be realized, 
under ideal conditions, from the heat required to elevate the tem- 
perature of unit mass of water from T 2 to T\, 



T t rp__rp 



J1 T 



Tr-Tz-TzlogpJ. 



The work that, under ideal conditions, could be realized from the 
heat taken from the source during vaporization, since this is 
absorbed at constant temperature, is 

TF " =/n ^TT ?; ••••••• (8) 

where n is the heat of vaporization at the temperature T\. 

Adding equations (7) and (8), we obtain for the total work 
that may be realized, for the given conditions, 

Wi = W' + W"=j(T 1 -T 2 -T 2 log^+r 1 ^^y . (9) 

Since the total heat, abstracted from the source, expressed in 
mechanical units, is 

J(T 1 -T 2 +r 1 ), 



124 THERMODYNAMICS 

the maximum work which would have been realized, had the 
operation been on a Carnot cycle, is 

W 2 = J(T 1 -T 2 +r 1 ) J ^^ (10) 

^ 1 

Dividing equation (9) by equation (10), we find 

W 2 T 1 +r 1 -T 2 { } 

If now, in equation (11), 

yr^Jrlog ^~>T 2 , (12) 

then W1/W2 is less than unity. To prove that the expression, 
given by the inequality (12), holds, we assume that T 2 , the tem- 
perature of the condenser, is fixed, and that T\ is a variable, which 
may be represented by T; remembering that T is always greater 
than T 2 , and that both are positive. Expression (12) may then 
be written 

TTo T 
^jr-\og-^==kT 2 ; (13) 

where k is a proportionality factor. From this, we find 

log^^(l-f) (14) 

Substituting, in equation (14), for T/T 2 , a new variable, x,we have 

log x = k( 1 

and, differentiating with respect to x, we obtain 

k =*=Y 2 , (15) 



STEAM AND STEAM ENGINES 125 

from which, if T/T2 equals unity, k equals unity; and, if T/T2 
becomes greater than unity, k must be greater than unity or 
equation (15) cannot hold. But this means that the left-hand 
member of equation (13) must be greater than T2', and hence 
W1/W2 is less than unity. It therefore follows that a steam 
engine, which rejects condensed steam to a boiler cannot, even 
under perfect conditions, convert into work as large a fractional 
part of the heat taken from the source as can an engine operating 
on a Carnot cycle, between the same limits of temperature. 

To illustrate the foregoing, we will deal with a concrete case; 
i.e., assume the temperature of the entering steam, and of the con- 
denser, respectively, 356°F. and 140°F. This gives: ri = 816,* 
72 = 600, and ri = 865. Substituting these values, in equation 
(11), we find 

ciaiqa* 816X60 0, 816 
Wl 816 + 865 -8i6^600 log 6 00 mn 
W 2 = 816+865-600 = 9L0 per Cent ' 

giving a loss of about 9 per cent due to not taking in all the heat 
at the maximum temperature. 

103. Unresisted Adiabatic Expansion of Steam. If dry 
saturated steam is allowed to expand adiabatically, from a chamber 
of given pressure to one of lower pressure, without doing work, 
the steam becomes superheated. This is due to the fact that, 
when the steam enters the chamber of lower pressure, eddy cur- 
rents are developed; and as they subside, the kinetic energy, 
possessed by them, is converted into heat. Since the process is 
adiabatic, and no external work is done, the total heat content, 
i.e., the total quantity of heat contained by the steam, will be 
the same at the end of the process as it was at the beginning. 
But since, the total heat of steam decreases as the pressure is 

* According to recent experiments, the zero for the thermodynamic scale 
is 491.65°F. below the melting-point of ice; but, in general, 492 is sufficiently 
accurate. 



126 THERMODYNAMICS 

decreased, and the final pressure is lower than the initial pressure, 
the steam must become superheated. 

If the steam is not initially dry, then it will become drier by 
unresisted adiabatic expansion. Assume that we are dealing with 
a unit mass of a mixture of steam and water under a pressure 
pi, for which the heat of the water and the heat of vaporization 
are, respectively, hi and T\. The total heat of the mixture, then is 

# = /*i+2iri; (16) 

where q\ is the dryness; i.e., the fractional part of the liquid which 
is present as steam. After expansion, since the total heat content 
remains the same, we have 

H = h 2 -\-q 2 r 2 ; (17) 

where h 2 , q 2 , and r 2 are, respectively, the heat of the liquid, the 
dryness, and the heat of vaporization for the final pressure p 2 . 

Equating the right-hand members of equations (16) and (17), 
we obtain 

hi+qm = h 2 -\-q 2 r 2 . . ... . . (18) 

By priming is meant the percentage of moisture present; and 
if this is low, the steam may become superheated by the unresisted 
adiabatic expansion, and q 2 , in equation (18), becomes unity. 
It will be shown later how, under certain conditions, advantage 
may be taken of this, and the initial priming determined exper- 
imentally. 

104. Resisted Adiabatic Expansion. If steam, the initial 
priming of which is low, expands adiabatically in such a manner 
that external work is done, it will become wetter. 

If in equation (9), Art. 102, it is assumed that complete 
evaporation has not taken place before the adiabatic expansion 
begins, then the work, expressed in heat units, yielded per cycle, is 

W=T 1 -T 2 -T 2 \ogp+qir 1 ?^^; . . (19) 



STEAM AND STEAM ENGINES 127 

where qi is the dryness. The total heat absorbed, in elevating 
the temperature of the water from T2 to T\, and evaporating it 
to the dryness qi, is 

Hi=Ti-T 2 +qiri (20) 

And since, under the assumed conditions, the difference between 
the heat abstracted from the source and that converted into 
work, must be equal to H r , the heat rejected to the condenser, 
we find, by subtracting equation (19) from equation (20), 

ff'=girig+!T 2 logg (21) 

But, the heat rejected to the condenser, after adiaoatic expansion 
to the temperature T2, must be equal to the heat liberated during 
condensation, i.e., 

H' = q 2 r 2 ; (22) 

where qi is the dryness and ? 2 the heat of vaporization corre- 
sponding to the temperature TV Equating the right-hand mem- 
bers of equations (21) and (22), we obtain 



from which 



T2 , m 1 ^1 
q 2 r 2 = qin ^r + T 2 log ^- J 

g2= ^fe +l0g rJ (23) 



Equation (23) enables us to compute the dryness, during 
resisted adiabatic expansion, provided the initial dryness be 
known. 

In the next chapter, the relation expressed in equation (23) 
will be deduced by a much simpler and shorter method. 



CHAPTER X 
ENTROPY 

105. It is obvious that a substance, in going from one isotherm 
to another, always suffers the same definite change in temperature; 
and furthermore, that this change in temperature is independent 
of changes in pressure and volume. That is, the change in tem- 
perature in going from one isotherm to another is independent 
of the path pursued during the change. A good analogue of this 
is the change in potential a body undergoes in going from a sur- 
face of potential Vi, to a surface of potential V2, the change in 
potential, V2 — V\, being independent of the path pursued in 
bringing about the change. 

It will now be shown that, in going from one curve to another, 
both curves representing reversible adiabatic processes, there is 
some definite constant change. Equation (19) of Art. 48 specifies 
for reversible adiabatic processes 

C v dT+pdv = 0; ....... (1) 

from which, by substituting for p its value as obtained from the 
characteristic equation, 

pv = RT, 

and separating the variables, we find 

C.f+1%-0 (2) 

By integrating equation (2), between limits, we obtain 

C.log^+JKlogf = 0; (3) 

128 



ENTEOPY 129 

where T\ and v\ are, respectively, the temperature and volume 
before the change, and T and v, respectively, the temperature and 
volume after the change. Since equation (3) is equal to zero, 
no matter what the limits of integration, it follows that there is 
something which does not change during a reversible adiabatic 
process. Integrating equation (2) for the primitive, we find 

CAogT+R\ogv = k', (4) 

where k is a constant of integration. Equation (4) shows that 
the fundamental differential equation for a perfect gas, yields upon 
integration for a reversible adiabatic process a constant. But since 
T and v, in equation (4), may have any values whatsoever, pro- 
vided, always, they are so related that the process is adiabatic, 
it follows that, no matter what the range, there is some function 
which remainsconstant; which conclusion is the same as that drawn 
from equation (3). Hence, since there is some function which 
remains constant during a reversible adiabatic change, there must 
be some definite constant change in going from one curve, repre- 
senting a reversible adiabatic process, to another curve, represent- 
ing a reversible adiabatic process. 

In equation (4), the constant k evidently represents some 
particular condition for the gas, which remains constant, during 
an adiabatic process; and its value depends upon the unit of 
measure and zero chosen. The condition of a gas, as expressed 
by equation (4), was called by Clausius the entropy of the gas; 
and, as just stated, the numerical value of the entropy depends 
upon the units chosen and the arbitrary zero from which it is 
measured. 

Equation (4) may now be stated as follows: The entropy 
of a substance during a reversible adiabatic change remains 
constant. 

106. Change of Entropy. If the left-hand member of equation 
(1) is not equal to zero, i.e., heat is either added or abstracted 



130 THERMODYNAMICS 

while the gas changes in volume and temperature, the process is 
no longer adiabatic, and the equation becomes 

dQ = C v dT+pdv; 

from which, by substituting for p its value as obtained from the 
characteristic equation, we have 

dQ = C v dT+RT— (5) 

Dividing equation (5) by T, we obtain 
dQ dT dv 

If we represent the entropy of the gas by <p, equation (4) 
becomes 

<? = C v \ogT+R\ogv; 

and, if the entropy is variable, 

, n dT „dv .„. 

dy = C V Y+Rj (7) 

The right-hand members of equations (6) and (7) being equal, 
it follows that 

dv = Y> ( 8 ) 

i.e., for a reversible process, the change in entropy is numerically 
equal to the ratio of the change in heat to the temperature at which 
the change takes place; the temperature being measured on the 
thermodynamic scale. 

The foregoing may be illustrated by equation (18) of Art. 94, 
which states that for a Carnot cycle operating on a perfect gas, 

Qi-Q 2 = T 1 -T 2 
Qi T x ' 

from which 

*"■* » 



ENTROPY 



131 



Equation (9) shows that the change in entropy in going from 
the adiabatic AD, (Fig. 16), to the adiabatic BC, is the same whether 
the change takes place along the isotherm AB or DC; since the 
ratio of change in heat to the temperature at which the change 
takes place is the same in both cases. It is obvious that the same 
ratio holds for any other isotherm cutting the two adiabatics 
AD and BC. It is, however, not necessary that the change take 
place along an isotherm. For, assume as depicted in Fig. 16, 
the irregular path ef to be cut by the two adiabatics aa! and W , 
which differ by an indefinitely small interval. The change in 




v 
Fig. 16. 



entropy, in going from g to h, along the irregular path ef, may be 
resolved into the two component changes; i.e., the change in 
entropy, in going along the isotherm gi, which is dy = dQ/T; 
where dQ is the change in heat and T the temperature at which 
the change takes place. The other component ih, being adiabatic, 
involves no change in entropy; hence, the change in entropy, 
in going from g to h, is 

dQ 

T' 



dy 



But since, as has just been shown, the change in entropy, in going 
from one adiabatic to another is the same for all isotherms, it 
follows, since gi is an isotherm, that the change in entropy in 



132 THERMODYNAMICS 

going from g to h is equal to the change in entropy in going from 
a to b along the isotherm Ti, and also to the change in entropy in 
going from a' to b' along the isotherm TV Similarly, it can be 
shown that the change in entropy in going along the irregular 
path ef, between any two adiabatics, is equal to the change in 
entropy in going between the same two adiabatics along either the 
isotherm T\ or TV It therefore follows that the change in entropy 
in going from the adiabatic AD, to the adiabatic BC, is always 
the same and is independent of the path by means of which the 
change is brought about. 

The foregoing demonstrations establish the fact that we are 
justified in making the assumption that there is a constant definite 
change in going from one reversible adiabatic to another; and, 
this being the case, it follows that during a reversible adiabatic 
process, some function, which has been termed entropy, must 
remain constant. It also follows that, since for a Carnot cycle 

Tx T 2 ' 

the source suffers a diminution of entropy during a cycle, which 
is precisely equal in amount to the entropy gained by the refrig- 
erator. 

107. Universal Increment of Entropy. The conduction of 
heat, such as discussed in Art. 77, is an irreversible process; and 
if, during a given interval of time, a quantity of heat Q is abstracted 
from a source at the temperature Ti, then, if steady conditions 
have been assumed by the prism, an equal quantity of heat will 
be rejected during the same interval of time, to some receiver at a 
lower temperature, say Ti. The loss in entropy, of the source, 
is then 

Q. 



ENTROPY 133 

and the gain in entropy of the receiver is 

. - Q 
92 - ¥ 2 - 

The gain in entropy of the system, since that of the prism is 
unchanged, is 

„ n _Q Q Q(T 1 -T 2 ) . 

n-n- ¥2 - ¥i - TiTi (10) 

Since the right-hand member of equation (10) is positive, it fol- 
lows that the entropy of the system, due to conduction, has 
increased; and further, since the work which would have been 
realized on a Carnot cycle, for the quantity of heat Q, oper- 
ating between the same temperature limits, is 

Q Tl ~ T2 

we see that the work, expressed in heat units, which has been 
irrevocably lost, due to the quantity of heat Q being transferred 
by conduction from the temperature T\ to T2, is numerically 
equal to the product of change in entropy and temperature of the 
receiver. 

We will now consider this in a wider sense. Assume, first, 
an engine working direct, which is thermodynamically perfect; 
i.e., one which maintains, during its operation, the sum of the 
entropies of source and refrigerator constant. If now, the mechan- 
ism upon which the engine does work is perfect and capable at 
any time of restoring all the energy imparted to it, then the process 
is perfectly reversible. This, however, is never the case; since 
all processes are attended by friction and a consequent develop- 
ment of heat, which is imparted, by conduction and radiation, 
to the surrounding bodies, there is necessarily an increment in 
entropy. To put it still more broadly, since heat can be only 
partially converted into work, and all energy, by friction, ohmic 



134 THERMODYNAMICS 

resistance, hysteresis, impact, etc., is finally degenerated into heat, 
it would appear that the entropy of the Universe, such as we 
know it, is tending toward a maximum. And the most gener- 
alized definition we can give, is :* The change in entropy that a system 
undergoes during a given irreversible process is a measure of the 
irreversibility of the process. This is indicated, in a limited way, 
by equation (10). 

108. The concept of entropy has been here introduced, not 
on account of its great scientific value, in the domain of theo- 
retical physics, but rather because so many of the discussions 
of practical thermodynamics are simplified so largely by its use. 
For our purposes, the two most important statements are: The 
change in entropy during a reversible process, is numerically equal 
to the ratio of change in heat to the temperature at which the change 
takes place; and reversible adiabatic processes are also isoentropic. 

109. Temperature Entropy Diagrams. From the equation 



-J: 



dQ 
T' 



where 9 is the change in entropy, it follows, immediately, that 

dQ = Tdr, 
and, for a reversible isothermal process, we have 

Q=rjj^=ru 2 - 91 ) (ii) 

Applying equation (11) to a Carnot cycle, we have for the heat 
abstracted from the source, during isothermal expansion, 

Qi = ri(q>2-<pi); (12) 

* For a comprehensive discussion of entropy, see Planck's "Thermo- 
dynamik," and also "Acht Vorlesungen uber Theoretische Physik," by the 
same author. 



ENTROPY 



135 



and during isothermal compression, for the heat rejected to the 
refrigerator, 

Q 2 = T 2 (n-n) (13) 



From equations (12) and (13), we find 

Q1-Q2 T x -T 2 



Q] 



T! • 



(14) 



The foregoing may, conveniently, be represented diagram- 
matically, by plotting the T-q* (temperature-entropy) diagram; 
using temperatures as ordinates and entropies as abscissas. Let, 



A 




T, 


B 


', 






\ 


D 




T 2 


C 








F 




E 



Fig. 17. 

in Fig. 17, the point A represent the condition of the working sub- 
stance, as regards temperature and entropy, at the instant it 
is put into contact with the source. Expansion now taking place 
at constant temperature, together with the absorption of the 
quantity of heat Qi from the source, the entropy increases by an 
amount 



92— 91 = jT> 



(15) 



and, the condition of the working substance, as regards temperature 
and entropy, is represented by the point B. Since, during this 
change, the temperature is constant, the line representing the 
change in entropy is a straight line parallel to the <p axis. Further- 



136 THERMODYNAMICS 

more, since during the adiabatic expansion, the entropy of the 
substance remains constant and only the temperature varies, 
this change is represented by the line BC parallel to the T axis; 
where the point C represents the condition of the substance, as 
regards temperature and entropy, when put into contact with 
the refrigerator at the temperature TV Compression now taking 
place at constant temperature, together with the rejection of the 
quantity of heat Q 2} to the refrigerator, the entropy decreases by 
an amount 

92-9i = ||; ......... (16) 

and the condition of the working substance, as regards temper- 
ature and entropy, is represented by the point D. This change 
in entropy is represented by the line CD parallel to the <p axis. 
Finally, during adiabatic compression, the temperature rises from 
T 2 to T\\ this change being represented by the line DA parallel 
to the T axis. 

From equations (15) and (16) we find 

Qi-Q2 _Ti-T 2 
Qi T 1 > 

as before, and the Carnot cycle on the T- 9 diagram is represented 
by a rectangle; the heat abstracted from the source, during a 
cycle, being measured by the area 

FABE=T 1 (y 2 -n), 
and the heat rejected to the refrigerator is measured by the area 

FDCE=T 2 (n-n)- 

The difference between these two areas is a measure of the heat 
converted into work; i.e., the area 

ABCD = (T 1 -T 2 )(n~n) 
is a measure of the external work done. 



CHAPTER XI 
APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS 

110. In Art. 102, it was shown analytically that, even under 
perfect conditions, a steam engine and boiler cannot convert into 
work as large a fractional part of the heat taken from a source as 
can an engine operating on a Carnot cycle. We will now show 
this by means of the T- <p diagram. As a matter of convenience 
it will be assumed that we are dealing with a unit mass of water, 
and, furthermore, that its thermal capacity is constant and equal 
to unity, between the temperature of the condenser and boiler. 
Under these conditions, the entropy, per unit mass of water, for 
any temperature T, is 



-4 



r.T =lo8 3V (1) 



where To is the temperature corresponding to the condition from 
which the entropy is measured, and c equals unity. Hence, we 
have a logarithmic curve instead of a straight line for one of the 
sides of the T- <p diagram. Let, as in Fig. 18, the point D represent 
the condition of the water, as regards temperature and entropy, 
when it is returned to the boiler. The logarithmic curve DA, 
then represents the change in entropy with respect to change in 
temperature as the water is heated from the temperature T2, 
that of the condenser, to the temperature T\, that of the boiler. 
The water is now evaporated at the temperature T\, and its change 
in entropy is represented by the line A B. From the point B, 
this point representing the condition of the water, as regards 
temperature and entropy, for complete evaporation, the steam 

137 



138 



THERMODYNAMICS 



expands adiabatically, its entropy remaining constant, until the 
temperature has fallen to T2, that of the condenser; its condition 
being now represented by the point C. The steam is now com- 
pressed isothermally, at the temperature T2, until complete con- 
densation has taken place, and the initial condition, represented 
by the point D, is reached. 

During the cycle, just described, the net work done, or the 
heat converted into work, is measured by the area ABCD; and 



G r A 


Tx I 


J 




T 2 


C 

E 


f; 



Fig. 18. 

the heat abstracted from the source, is measured by the area 
FDABE. Hence, the ideal coefficient of conversion is 



Y) = 



Area ABCD 
Area FDABE' 



(2) 



Had the process been a Carnot cycle, the ideal coefficient of 
conversion would be expressed by 



Area DGBC 
Area FGBE 



(3) 



The value of yj as given by equation (2) is obviously less than that 
given by equation (3); which is in agreement with the results 
obtained in Art. 102. 



APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS 139 

111. Change of Dryness during Adiabatic Changes. Equation 
(23), of Art. 104, may be found in a very simple manner from the 
fact that the entropy of a substance, during reversible adiabatic 
changes, remains constant. The gain in entropy, for a unit mass 
of water, in being heated from a temperature To to the tempera- 
ture T\ is, by equation (1), 

T 
<pi = log^; (4) 

and the gain in entropy in evaporating to a dryness qi, at the 
temperature T\, is 

<P2 = giyr; ....... (5) 

where r\ is the heat of vaporization corresponding to the tem- 
perature T\. Taking the sum of equations (4) and (5), we find, 
for the total change in entropy 

9=9i+92 = log^r+giyr (6) 

But for reversible adiabatic changes the entropy remains constant ; 
hence, if, after the condition expressed by equation (6) has been 
attained, the temperature due to an adiabatic change, which may 
be either expansion or compression, changes to T2, and the dryness 
changes to #2, we must have 

T\ n T2 T2 

log ^+21^ = ^-^+22^; .... (7) 

where V2 is the heat of vaporization corresponding to the temper- 
ature TV From equation (7), we find 

which is the same as equation (23) of Art. 104. 

112. Dryness by Means of Temperature-Entropy Diagram. 
The change in dryness which a mixture of water and steam, or 



140 



THERMODYNAMICS 



any liquid and its vapor, undergoes during adiabatic changes, 
provided the initial dryness be known, may readily be found by 
means of the T-qp diagram. Let, in Fig. 19, the curve DA be 

T 

plotted, to proper scale, using various values of <p = log — as abscis- 

To 

sas and the corresponding values of T as ordinates; then this 
curve represents the relation of temperature and entropy for 





A 


Tx 


E B 




m/ 




T 


nj_ 


\° 






To 





Fig. 19. 

unit mass of water between the temperatures To and T\. If now, 
for various points along the curve DA, between To and T\, hor- 
izontal distances be measured off towards the right, each distance 
being equal to r/T, where T is the temperature corresponding 
to the point, and r the corresponding heat of vaporization, the 
curve of saturation BC is found. By construction, then, at any 
temperature T, the horizontal distance between the curves AD 
and BC represents the increase in entropy due to the heat added 
to bring about complete evaporation at that temperature. Thus, 
for the temperature T, we have 



= - = mo) 



APPLICATIONS OF TEMPEEATURE-ENTEOPY DIAGRAMS 141 



TEMP. DEGREES FAHR. 

g S? 8 

© o o 











































































































































































































































































































































































































































































































£U 


£r 




















































































































































































































































































































































































































































































RELATION OF 

TEMPERATURE AND ENTROPY 

FOR WATER AND 

SATURATED STEAM 


































































































































































































































































































































































































































































































































































- 


-e- 

< 

+ 




































































































































































































































































































































































































































































































































































































































































































































































































































































































































^- 








































































W* 


j.~ 

























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































142 THERMODYNAMICS 

where r is the heat of vaporization corresponding to the temper- 
ature T, and <p the increment in entropy. 

Let, now, the initial dryness, at the temperature T\, be qi, such 
that the entropy is represented by the point E. Since the quantity 
of liquid evaporated, at a given temperature, is directly propor- 
tional to the quantity of heat added, and the increment in entropy 
is also directly proportional to the quantity of heat added, it 
follows that the increment in entropy is directly proportional 
to the amount of evaporation. Hence, the initial dryness is 



AE 
qi= AB' 



If, now, adiabatic expansion take place, the entropy remains 
constant, and the vertical line EF represents the relation of tem- 
perature and entropy. Hence, the dryness corresponding to the 
temperature T is given by 

«Z = ^ (8) 

mo 

In a similar manner, provided always the initial dryness be 
known, the dryness corresponding to any temperature during 
an adiabatic change, may be found. And it makes no difference 
whether we are dealing with an expansion or a compression. 

It is obvious that if a T- 9 curve be plotted to a convenient 
scale, for water, together with the corresponding saturation curve, 
in a manner as has just been described, between such temperature 
limits as are likely to occur in practice, we may at once, from such 
a sheet, provided always the initial dryness be known, determine 
the amount of dryness at any temperature for adiabatic changes. 
As a matter of convenience, such a sheet is given on page 141. 

113. Zero Curve. Assume that various horizontal distances 
between the curves AD and BC are all divided into the same 
number of parts, and each part is the same fractional part of the 



APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS 143 



total distance. If the points, so found, are connected by smooth 
curves, as shown in Fig. 20, then the dryness along any particular 
curve is a constant. For, the horizontal distance between any 
curve, such as mn, and the curve AD, no matter at what temperature 
the distance be measured, is always the same fractional part of 
the total increment in entropy, at that temperature, due to vapor- 
ization; and therefore, represents the same fractional part of 
vaporization. If, now, an adiabatic, such as ab, be drawn, it is 
found to cut the curve mn; i.e., the curve mn passes to the right 
of the adiabatic ab, as the expansion progresses, and shows that 



A co Tj ma B 





pd 



bn 



Fig. 20. 

the steam becomes wetter during adiabatic expansion and drier 
during adiabatic compression. If, on the other hand, an adiabatic, 
such as cd, be drawn, it is found that the curve of equal dryness 
op, as the expansion progresses, passes to the left of it; and shows 
that the vapor becomes drier during adiabatic expansion and wetter 
during adiabatic compression. It is thus seen that if the vapor 
be initially quite dry, it becomes wetter during adiabatic expansion, 
and if the dryness is very low, it becomes drier when expanded 
adiabatically. 

If the adiabatic at any point becomes tangent to the curve of 
constant dryness, then there is at that point, no change in dryness 
during adiabatic changes. By finding a number of points on the 
various curves where the tangent is vertical, and joining these 



144 THERMODYNAMICS 

points by a smooth curve, then at any point to the right of this 
curve the dryness is decreased by adiabatic expansion, and at 
any point to the left of this curve, adiabatic expansion increases 
the dryness. Such a curve is known as the u zero curve" ; and 
for temperatures such as are common in practice, does not lie 
very far from 50 per cent dryness. 

114. Loss of Work Due to Using Steam Non-expansively. 
Assume, in the first place, that no expansion whatsoever is allowed; 
but that at the instant of cut-off, the steam is put into contact 
with the refrigerator, and condensation takes place at constant 
volume. Then the work lost, due to using the steam non-expan- 
sively, is represented by the area BCK, of Fig. 15, for the ideal 
case discussed in Art. 102. 

As a matter of fact, during the greater part of the eighteenth 
century, steam was used in this manner; i.e., the steam was not 
worked expansively, but immediately after cut-off, the steam 
was condensed by a jet of water, .either in the cylinder, or in an 
adjoining condenser. The result, however, is the same whether 
the steam be condensed by a jet of water, immediately after cut-off, 
or allowed to escape to a space of lower pressure; for in either case, 
there is the same gradual diminution of pressure in the cylinder. 

The conditions which obtain, when steam is used non-expan- 
sively, are best studied by the aid of the T- <p diagram. In 
Fig. 21, DA is the !F-<p curve for the heating of unit mass of 
water, AB the curve for evaporation at the temperature T\ t 
BC the saturation curve, BE the curve for adiabatic expansion, 
BnF the curve of condensation at constant volume, and FD the 
condensation curve at constant temperature. The condensation 
curve BnF is determined as follows: For any temperature T, we 
have to determine a point n such that 

mn 
q = — ; 
mp 

where q is the dryness corresponding to the temperature T. 



APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS 145 



The volume of the steam in the cylinder, remaining sensibly 
constant, since the volume of liquid present is practically negli- 
gible, we have 

qs = si; (9) 

where s is the volume of unit mass of saturated steam at the tem- 
perature T, and s\ the volume originally occupied at the temper- 
ature T\. s and si, are found from steam tables. From equation 
(9) we find 



Q = 



Sl 



(10) 



A 




T, 


B 


/ 






^!\ 


m/ - 


T 


r Ls^^ 


°' \p 


/ 






I \ 


/ 

D 


?/ 




E l Ac 




r 2 


■*- -*L> 











Fig. 21. 



But, as has been previously shown, a relation must subsist, such 
that 

mn 



q= 



mp 



(id 



hence, by combining equations (10) and (11), we find 



mn = mp—. 
s 



(12) 



Finding a number of points in this manner, the curve BnF 
is determined; and the loss of work, due to using the steam non 
expansively, is obviously measured by the area BFE. 

115. Loss of Work Due to Incomplete Expansion. If there 
is a partial adiabatic expansion before exhaust or condensation, 



146 



THEKMODYNAMICS 



then the T-y diagram takes the form as depicted in Fig. 22. 
DA is the T- 9 curve for the heating of unit mass of water, AB 
the curve for complete evaporation, at the temperature Ti, BC 
the saturation curve, BG the curve- for adiabatic expansion, to 
the temperature V ', GnF the curve of condensation at constant 
volume, and FD the curve of condensation at the temperature 
T2. To determine the curve of condensation at constant volume, 
we must find a point n for the temperature T, such that 



mn 

q= — ; 

mp 



(13) 



A 




T, 


B 




„/ 




T' 


G 


V 




/ 










/ 


T 2 


_T_ 


/ / 

/ 

/ 


a. 


\P 


D 




F 


b E 


C 



Fig. 22. 

where q is the dryness corresponding to that temperature. Now, 
the volume occupied at the point G, corresponding to the tempera- 
ture T', is 

where q' is the dryness at the temperature T', and s' the volume 
of unit mass of saturated vapor at T' . Also, for constant volume ! 



qs = q's'\ 



(14) 



where q is the dryness at the temperature T, and s the correspond- 
ing volume for unit mass of saturated vapor. From equation 
(14), we have 

ff-fff (15) 



APPLICATIONS OF TEMPEEATUEE-ENTEOPY DIAGEAMS 147 
Finally, combining equations (13) and (15), we obtain 



s 
mn = q f -mp (16) 

o 



Finding a number of points in this manner, the curve GnF is 
determined. If the steam is initially not dry, the curve GnF is 
determined in precisely the same manner; but, the curve BG is 
shifted toward the left by a fractional part of the length AB, 
depending upon the amount of initial priming. 

The work lost, due to incomplete expansion, is measured by 
the area GFE; and, by an inspection of the figure, it becomes 
obvious that the loss of work decreases very rapidly as the expan- 
sion is increased. As an example, were the expansion continued 
up to the point a, the loss of work, due to incomplete expansion, 
would be measured by the small area abE. 

The greater the amount of expansion, after cut-off, the longer, 
necessarily, the stroke of the piston; but, the longer the stroke, 
other things being equal, the higher the first cost of the engine, 
and the greater the loss of work due to friction. Hence, there 
must be a point beyond which it is uneconomical to carry the 
expansion. Besides increasing the friction, there are still other 
losses introduced, by carrying the expansion too far; these will be 
considered later. Just how far to carry the expansion so as to give 
the best economy is a problem far too complex to be solved theo- 
retically. At best, theory can only serve as a guide, and the most 
economical expansion must be determined experimentally. For 
a simple engine, the point of cut-off may vary from about one-third 
to one-sixth of the total stroke; depending upon whether the engine 
is running non-condensing or condensing. But, it must always 
be remembered that the ratio of cut-off to length of stroke depends 
upon various conditions, which will be better understood after 
we have dealt with the actual behavior of the steam in passing 
through the cylinder. 



148 



THERMODYNAMICS 



116. Gain of Work Due to Superheating. If the steam, 
after being completely evaporated, be superheated, the ideal 
coefficient of conversion is increased. But, this must not be under- 
stood to mean the same proportional gain in work; for, lubrica- 
tion and packing become more difficult as the temperature is 
increased; and when the temperature becomes very high, radiation 
becomes excessive. 

Let, in Fig. 23, DA be the T- <p curve for the heating of unit 
mass of water, AB the curve for evaporation, BE the curve for 









/ 


E 




A 


"ft 


./ 








1 
l\ 
n 










i 
i 


\ 
\ 






T 2 


h 


|D 






1 

1 




IK 






i 


F 



9 
Fig. 23. 



superheating, EI the adiabatic expansion curve, and ID the 
curve of condensation. The curve BE, for superheating, is found 
as follows: The thermal capacity of superheated steam, for 
temperatures such as are found in practice, is approximately 
constant; hence, we have 

C Ts dT 

from which 



cp = clog 



TV 



(17) 



APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS 149 

where <p is the change in entropy in going from B to E, T s the 
temperature to which the steam is superheated, and c the thermal 
capacity per unit mass of superheated steam. The thermal 
capacity per unit mass of superheated steam is, as has just been 
stated, practically constant, and is approximately equal to 0.480. 
With no superheating, and expansion along the adiabatic 
BJ, the ideal coefficient of conversion is, 

, ABJD 



ABGKD' 

and, with superheating and adiabatic expansion along EI, we 
find, for the ideal coefficient of conversion, 

„ ABEID 
^ ABEFKD' 

By an inspection of the figure, it becomes obvious that tq"> y/. 

If the steam be superheated to a temperature, such that the 
adiabatic EF passes through the point C, then the steam will 
be just saturated after it has been expanded to the temperature 
7 7 2. To find the amount of superheating that will bring about 
this condition, it is only necessary to equate entropies, for the 
points E and C. The change in entropy, in going from D to E, is 

and the change in entropy, in going from D to C, is 

* =T- 2 ' 

But, in order that the adiabatic EF pass through the point C, 
•p' must equal <p"; hence, 

log y 2 + t\ +c g Yi = Y 2 ' 



150 THERMODYNAMICS 

V 



from which 



losTs= ci¥ 2 ~¥ 1 ~ hs ^' 



By means of equation (18), T s is readily found. 

117. Double-acting Engine. Up to the present, we have been 
considering matters as though the engine were only single-acting; 
i.e., admission and exhaust take place only at one end of the 
cylinder. In general, however, this is not the case. By having 
proper valve arrangements, admission and expansion take place 
in one end of the cylinder while release and exhaust take place 
in the other end. And, the engine is double-acting; thus prac- 
tically doubling the capacity of the cylinder and giving a more 
uniform distribution of the work for a rotation of the fly-wheel. 
Hence, a single-acting engine, to carry its load properly, requires 
a fly-wheel of greater inertia than does a double-acting engine. 

118. Condensing Engine. By a condensing engine is meant 
an engine which exhausts to a receptacle of some kind, called a 
condenser, where the steam is condensed at a comparatively low 
temperature, and the pressure in the condenser is maintained 
constant and lower than that of the -atmosphere by means of a 
vacuum pump. In good condensers, the pressure is as low as 
the equivalent of one inch of mercury. By a non-condensing 
engine is meant an engine which exhausts directly to the atmos- 
phere at practically atmospheric pressure. 

Experience shows that, in general, when the supplied steam 
has a pressure of 100 lbs., or over, despite the fact that a certain 
amount of power is consumed in operating the vacuum pump, 
there is a decided gain in economy, when engines are operated 
condensing. Hence, in general, condensing engines are employed. 



CHAPTER XII 

ELEMENTARY STEAM AND ENGINE TESTS 

119. Before proceeding to discuss the actual behavior of 
the steam as it passes through the cylinder of an engine, a brief 
description will be given of the methods pursued in determining 
the dryness of steam. It is obvious, from the discussions in the 
preceding chapter, that one of the essentials in studying the per- 
formance of a steam engine is a knowledge of the condition of 
the supplied steam. But, aside from the temperature-entropy 
diagram, if we wish to plot an adiabatic for steam on the p-v 
diagram, we must know the initial dryness. 

According to calculations by Zeuner, the equation 

pv n = k, , (1) 

where k is a constant, may be used for adiabatic changes. But 
the value of n depends upon the initial dryness. This value 
of n is given by the empirical equation 

n = 1.035+0.1g; . (2) 

where q is the initial dryness. Equation (2) holds for all values 
between 70 per cent and 100 per cent dryness. Hence, if an 
adiabatic for steam is to be plotted, that value of n must be 
used in equation (1), which is found by means of equation (2), for 
the given initial dryness. 

120. Throttling Calorimeter. It was shown, in Art. 103, 
that steam becomes drier, during unresisted adiabatic expansion; 
and furthermore, under proper conditions, if the initial priming 
be low, the steam may become superheated. Depending upon 

151 



152 



THERMODYNAMICS 







/Jo, 



this principle, Professor Peabody designed a calorimeter, by 
means of which the dryness of steam, provided the priming is 
low, may be determined. In Fig. 24, S represents the supply 
pipe, and A a vessel into which the steam expands; the rate of 
inflow being regulated by the valve Vi, and the outflow by the 
valve V2. Gi is a gauge, indicating the pressure of the steam 

in the supply pipe, G2 a second gauge, 
indicating the pressure of b the steam 
in the vessel^, and m a thermometer, 
indicating its temperature. The valves 
Vi and V2 are so regulated that the 
pressure in the vessel A is always con- 
siderably less than the pressure in the 
supply pipe. The vessel A is either 
well lagged with some non-conducting 
material, or else highly polished, to 
reduce radiation to a minimum. After 
the flow of steam has continued for 
some time, steady conditions will 
obtain; and, if the priming of the 
steam in the supply pipe is low, super- 
heating will take place in the vessel 
A , and the thermometer m will register 
a temperature higher than that corresponding to saturated 
steam, under a pressure as registered by the gauge G2. Let 
ti be the temperature of saturated steam corresponding to 
the pressure indicated by the gauge G2, and T2 the tempera- 
ture registered by the thermometer; then T2 — ti is the amount 
of superheating. The total heat of unit mass of steam then, 
in the vessel A, is 



U 



m v * 



Fig. 24. 



H' = ^i+n+c(T2-Ti); 



(3) 



where E.' is the total heat, hi the heat of the liquid, corresponding 
to the temperature ti, r\ the heat of vaporization for this tern- 



ELEMENTARY STEAM AND ENGINE TESTS 153 

perature, and c the thermal capacity per unit mass for superheated 

steam. The total heat, per unit mass, for the steam in the supply 

pipe, is 

H" = h+qr; (4) 

where H" is the total heat, h the heat of the liquid for the tem- 
perature t, which is the temperature of saturated steam for the 
pressure as registered by the gauge Gi, r the heat of vaporization 
for this temperature, and q the dryness of the steam in the 
supply pipe. For an adiabatic flow, however, the total heat 
for the two conditions is the same; hence, the right-hand mem- 
bers of equations (3) and (4) are equal, and we have 

h+qr = hi+ri+c(r2—'zi); 
from which 

_ fei+ri+c(T 2 -Ti)-fe . , 

0- ~ r W 

In order to obtain reliable results, by the method just described, 
the two gauges G\ and G2 must be accurately calibrated. On 
the other hand, a slight error in the thermometer does not 
appreciably alter the result; since the amount of superheating 
is necessarily small, the quantity of heat involved is small in 
comparison with the other quantities. But, the thermometer must 
be of sufficient accuracy so that we may be assured that there is 
superheating. 

The amount of moisture that may be removed by throttling 
depends, of course, upon the difference between the pressure of 
the steam in the supply pipe and the pressure in the chamber 
into which it expands. If the pressure in the chamber is equal 
to that of the atmosphere, and the pressure in the supply pipe 
is 100 lbs. per square inch, then the dryness must be about 96 
per cent so that all the moisture may be removed by throttling. 
If the pressure in the chamber be reduced by means of a con- 
denser, a greater amount of moisture may be removed. If the 



154 THERMODYNAMICS 

initial pressure of the steam be 150 lbs. per square inch, and 
the pressure in the chamber is atmospheric, then about 5 per 
cent priming may be removed by throttling. 

121. Condensing Calorimeter. The dryness of steam may 
also be determined by condensation. There are various methods 
which may be pursued; one is to have a vessel partially filled 
with water at some low temperature, and passing steam into 
this until some convenient rise in temperature has been attained. 
The quantities of water and steam are determined directly by 
weighing, the initial and final temperatures of the vessel are 
read from a thermometer immersed in it, and the initial temper- 
ature of the supplied steam is determined from the pressure, 
as indicated by means of a gauge attached to the supply pipe. 

Let M be the water equivalent of the vessel and contents, 
m the mass of the condensed steam, and ti and T2, respectively, 
the initial and final temperatures of the vessel and contents, 
then if q is the initial dryness and t the corresponding temperature 
of the supplied steam, we have 

M(t2— Ti)+m/&2 = mqr+mh; (6) 

where lfi2 is the heat of the liquid corresponding to the temperature 
T2, r and h, respectively, the heat of vaporization and the heat 
of the liquid, corresponding to the temperature t. From equa- 
tion (6) we find 

ilf(r 2 -Ti)-m(fr-ft 2 ) (7) 

mr 

Equation (7) was deduced on the assumption that there is no 
radiation during the progress of the experiment. A correction 
for radiation may, however, be applied by taking a curve of 
cooling for the vessel. Due to the fact, that the mass of the 
condensed steam is determined by a difference in weighing, and 
that this mass is necessarily small in comparison with the mass 
of liquid initially contained in the vessel, a serious error may 
be introduced by an inaccuracy in weighing. 



ELEMENTARY STEAM AND ENGINE TESTS 155 

A better method than the one just described, is that of 
passing the steam through a spiral tube, contained in a condenser 
through which there is maintained a continuous flow of water, 
in a manner such that the steam is completely condensed and 
reduced in temperature to that of the outflowing condensing 
water. After steady conditions obtain, the mass of steam con- 
densed, during a given interval of time, is determined by col- 
lecting it in a suitable vessel; and in a similar manner, by 
collecting in a separate vessel, the mass of water which passes 
through the condenser, during the same interval of time, is 
determined. Knowing the initial and final temperatures of the 
condensing water, together with its mass, the mass of the con- 
densed steam and its initial and final temperatures, then the 
dryness of the steam is determinate. Let M and m, respectively, 
be the mass of the condensing water and condensed steam for the 
same interval of time, ti and 12, respectively, the temperature 
of the condensing water for inflow and outflow, and t the initial 
temperature of the supplied steam, then, since the temperature 
of the outflow and that of the condensed steam is the same, we 
have 

M(r2 — ii)^rmh2 = mqr-\-mh; (8) 

where h,2 is the heat of the liquid corresponding to the temper- 
ature T2, r and h, respectively, the heat of vaporization and the 
heat of the liquid corresponding to the temperature t, and q 
the dryness. From equation (8), we find 

mr 

Equation (9) was deduced on the assumption that there is no 
radiation during the progress of the experiment. If the tem- 
perature of the vessel differs materially from that of the sur- 
roundings, corrections for radiation must be applied for this 
difference. Radiation may, however, be completely eliminated 



156 THERMODYNAMICS 

by regulating the inflow such that the vessel and contents are 
continuously at room temperature. 

122. Separating Calorimeter. Professor Carpenter devised an 
apparatus by means of which the moisture, present in the steam, 
is removed mechanically. The steam is passed from the supply 
pipe into a chamber, where it strikes against a convex surface, 
surrounded by a wire mesh, through which the escaping steam 
must pass. When the steam strikes the cup, the water present 
is separated, passes through the mesh, and is collected in the 
chamber; the dry saturated steam, meanwhile, passes into an 
outer jacket, which surrounds the chamber, and escapes from 
an orifice at the bottom, where it is condensed and collected. 
The quantity of water collected in the chamber is read directly 
from a glass gauge, which has been previously calibrated; and the 
quantity of steam which passes through the calorimeter is deter- 
mined by condensing and weighing. 

In making a determination, the valve in the pipe supplying 
steam to the chamber is opened, and when steady conditions 
obtain, a reading is taken on the glass gauge, and simultaneously, 
.the exhaust pipe is passed into the condenser. When the oper- 
ation has been continued for a sufficient interval of time, the 
gauge is again read, and at the same instant, the exhaust pipe 
is removed from the condenser. The mass of steam, passing 
through the apparatus for the given interval of time, is found 
directly by the difference in weight of the condenser for final 
and initial conditions. And this mass compared with the sum 
of the two masses, i.e., the mass of the condensed steam and the 
mass of the water collected in the chamber, gives the dryness. 

One of the inherent difficulties common to all methods, in deter- 
mining the dryness of steam, lies in obtaining a sample which is 
a fair average of the steam supplied to an engine. 

123. Clearance. The volume swept out by the piston of an 
engine, during a stroke, is equal to the product of the area of 
the piston and length of its stroke. The volume between the 



ELEMENTARY STEAM AND ENGINE TESTS 157 

piston and cylinder head at the end of the stroke, plus the 
volume of the supply and exhaust-passages leading to the admis- 
sion and exhaust-valves, is called the clearance. The clearance 
then is, that part of the volume through which the piston does 
not sweep, and is readily found by closing the valves and 
determining the volume of water required to fill the space when 
the piston is at the end of its stroke. A convenient way of 
expressing the clearance of an engine is by a ratio; i.e., the ratio 
of the volume of the clearance, to the volume of piston displace- 
ment plus volume of clearance. The ratio of the clearance 
volume to the piston area gives the equivalent length of 
clearance. 

The clearance of different engines varies considerably, depend- 
ing upon the size of engine and type of valves used; and, other 
things being equal, the clearance for small engines is relatively 
larger than it is for large ones. In practice, depending on the 
type of engine, the clearance may vary from 2 per cent to 10 per 
cent. 

124. Cushion Steam and Cylinder Feed. The mass of steam 
which remains in the clearance-space at the end of the exhaust- 
stroke, depends upon the time of closing of the exhaust-valve. 
Thus, if the exhaust-valve does not close until the exhaust- 
stroke has been completed, then the pressure of the steam, 
remaining in the cylinder, is the same as that of the condenser, 
and the mass of the steam is equal to the product of the clearance- 
volume and the density of the steam. If, on the other hand, 
the exhaust-valve closes before the exhaust-stroke has been com- 
pleted, then the pressure of the steam, remaining in the cylinder 
at the end of the exhaust-stroke, will be higher than that existing 
in the condenser; hence, in this case, the mass of steam remaining 
in the cylinder is greater than that for a later closing of the 
exhaust-valve. The steam remaining in the cylinder, at the end 
of the exhaust-stroke, is called the cushion steam; and the steam 
drawn from the boiler, per stroke, is called the cylinder feed. 



158 THERMODYNAMICS 

During expansion, both quantities are present, whereas, during 
compression, the cushion steam alone is present. 

125. Wire Drawing. If the exhaust-valve closes late, the 
pressure of the cushion steam is less than that of the steam in 
the supply pipe, and a certain quantity of steam must pass 
into the cylinder, during each stroke, before the maximum 
pressure is reached. The entering steam, therefore, does not 
do as great an amount of work on the piston as it would do if 
the cushion steam had been compressed to the pressure of the 
incoming steam; in other words, it is a case of imperfectly 
resisted expansion. It is true that the incoming steam, if dry, 
becomes superheated, and if partially wet becomes drier, due 
to the partially unresisted expansion; but the pressure being 
lower, the heat which is evolved when the eddy currents subside, 
is applied at a lower temperature, and therefore, the imperfectly 
resisted expansion constitutes a thermodynamic drop. 

If the exhaust-valve closes at the proper time, then the 
pressure of the cushion steam is equal to the pressure of the 
incoming steam, and the thermodynamic drop, so far as this 
part of the action of the engine is concerned, is avoided. Fur- 
thermore, the work which is done on the cushion steam, in 
compressing it from the condenser pressure to that of the incom- 
ing steam, is precisely equal to the work done by it, in expanding 
between the same limits of pressure; hence, there is no loss of work 
involved due to compression. There are, however, other unavoid- 
able losses. The piston advances rapidly, calling for a large sup- 
ply of steam, and the admission-valve does not open instantan- 
eously, but requires a definite time interval. Hence, due to the 
resistance offered to the flow of steam, by the supply passages 
and valves, there is a certain amount of throttling, the same as 
when the cushion steam is at a pressure lower than that of the 
supplied steam; causing the pressure in the cylinder, during 
admission, to be less than that existing in the supply pipe. And 
furthermore, in general, due to throttling, the pressure in the 



ELEMENTAEY STEAM AND ENGINE TESTS 159 

cylinder gradually decreases as the admission advances. The 
result of these combined causes, due to which the pressure in the 
cylinder during admission is lower than that of the supply pipe, 
or boiler, is known as wire drawing, and constitutes a thermo- 
dynamic drop. 

There is also a loss of work during the exhaust-stroke, due 
to the fact that the exhaust-passages and valves offer a resistance 
to the flow of steam, which makes the pressure in the cylinder, 
during exhaust, always higher than that of the condenser. 

126. The Indicator. One of the most important and, at the 
same time, one of the most delicate pieces of apparatus used in 
engine testing is the indicator. The indicator consists essentially 
of two parts ; the first part being a small piston P fitted accurately 
into a cylinder and controlled by a helical spring S. The spring 
may be either inside of the cylinder or, as shown diagrammatically 
in Fig. 25, outside. The type of indicators having the spring 
above the cylinder are more convenient; and furthermore, since 
the springs, in this form, are not subjected to the same fluctua- 
tions of temperature, the results obtained are more satisfactory. 
The cylinder of the indicator, by means of a short supply pipe 
containing a cock, is tapped onto the cylinder of the engine, 
over the clearance space, in a manner such that the steam in 
the engine cylinder exerts its full pressure against the piston 
of the indicator throughout the entire cycle. If the indicator pis- 
ton moves freely, i.e., without appreciable friction, and the spring 
obeys Hooke's Law, then the movement of the piston will be 
proportional to the fluctuations of the pressure in the cylinder 
of the engine. To magnify the motion of the piston of the indi- 
cator, the end of its piston rod is connected, by means of a system 
of links, to a lever, in a manner such that a pencil point p, carried 
by the end of this lever describes, between the limits of travel, 
practically a right line. The springs are accurately calibrated 
to a definite scale with respect to the motion of the pencil point. 
Thus, if a spring is a 60 lb. spring, it means that the pencil 



160 



THERMODYNAMICS 



point moves over a distance of 1 inch for a change in pressure, on 
the piston, of 60 lbs. per square inch; and a distance of 0.75 inches 
for a change in pressure of 45 lbs. per square inch, etc. 

The second part of the indicator consists of a drum D, 
controlled by a spring, upon which the indicator card is wrapped. 
The drum has wrapped around its lower part a cord C, which in 




Fig. 25. 



turn is connected by means of some mechanism, to the cross- 
head of the engine, in a manner such that the angular displace- 
ment of the drum is proportional to the linear displacement 
of the piston of the engine. The linear displacement of the 
surface of the drum, however, is less than that of the cross-head; 
i.e., the motion is reduced by the mechanism through which the 



ELEMENTARY STEAM AND ENGINE TESTS 161 

cord, operating the drum, is connected to the cross-head. From 
the foregoing, it is obvious that, if the drum is stationary, and 
the cock in the supply pipe between the cylinder of the engine 
and the cylinder of the indicator is open, the pencil point traces 
a straight line on the indicator card. On the other hand, if the 
stop-cock is closed and the drum is in motion, the pencil traces 
a straight line at right angles to the former. This line is the 
atmospheric line, since the stop-cock is so arranged that when 
the steam is cut off from the indicator cylinder, a vent opens, 
allowing free access of the atmosphere to the space below the piston 
of the indicator. If, however, the stop-cock is open, and the 
drum is moving in unison with the piston of the engine, the 
position of the pencil point of the indicator, at any part of the 
cycle, is a measure of the pressure and volume, of the working 
substance ; for that instant. Hence, during a cycle, the pencil 
point traces out a diagram, which shows to a reduced scale, as 
regards pressure and volume, the condition of the working sub- 
stance, for every part of the cycle. The diagram so traced, is 
the actual indicator diagram of the engine. 

127. Indicator Diagram and Valve Adjustment. By means 
of the indicator diagram, the behavior of the working substance 
may be conveniently studied for the entire cycle; and further- 
more, we are enabled by it to judge, whether or not, the valves 
are properly adjusted, which is very important; since any faulty 
valve adjustment may seriously affect the efficiency of the engine. 
Also, as will be shown in this chapter, by means of the indi- 
cator diagram, we are enabled to determine the power delivered, 
by the working substance, to the engine; hence, if the power 
delivered by the engine be known, the efficiency of the engine, 
as a mechanical contrivance, is immediately determined. 

Fig. 26 is a reproduction of an indicator diagram taken from 
one end of the cylinder of a 40-H.P. engine, making 300 r.p.m.; 
the engine working non-condensing; i.e., exhausting to the atmos- 
phere. Fig. 27 is the indicator diagram for the same end of 



162 



THERMODYNAMICS 



the cylinder when the engine was exhausting to a surface con- 
denser; a partial vacuum being maintained by a pump. 

AB is the admission line, BC is the expansion line, C being 
the point where the exhaust-valve begins to open, and D the 



H 


p 






I 


* 


A f 


V- 




B 










\^_ E 






AD 


F 


' 








G 



Fig. 26. 

point where it is fully open; DE is the exhaust line, and at 
the point E compression begins. Just how far the compression 
will be carried before the admission-valve opens depends upon 
the set of the valves. In the diagrams here shown, the admission- 




Fig. 27. 

valve opened at the point a. The various points being illy de- 
fined is due to the time element involved in the opening and closing 
of the valves. The. line OH is the line of zero volume, and is 
found by taking a distance, to the proper scale, to the left of 
FA, representing the equivalent length of the clearance. The 



ELEMENTARY STEAM AND ENGINE TESTS 163 

line of zero pressure, or vacuum line OJ, is found by measuring 
down from the atmospheric line FG, a distance representing 
the atmospheric pressure at the time the diagram was taken. 
Finally, HI shows, as registered by the gauge, the steam pressure 
in the supply pipe. 

128. Comparison of Theoretical and Actual Curves. If it be 
desired to compare the expansion or compression curve, with 
an isotherm or adiabatic, a point on the curve is chosen, preferably 
about the middle, and the theoretical curve is made to pass through 
this point. As a matter of convenience, the following method 
for plotting curves is here given. To plot the curve whose 
equation is 

pv n = k, 

we proceed as follows : In Fig. 28, OA is the line of zero volume, 
OB the line of zero pressure, and a a point on the curve. Lay 
off the line OC, making an angle with the line OA, and the 
line OD, making an angle a with the line OB, such that 

l+tan£ = (l+tana) n . ..... (10) 

Draw ad parallel to OB, and dh making an angle of 45° with OA. 
Now draw af parallel to OA, and through /, fg making an angle 
of 45° with OB; then the point b, which is the intersection of 
the line Kb, parallel to OB, with the line gb, parallel to OA, is a 
point on the curve. For, if we represent, for the point a, the pres- 
sure and volume respectively, by pi and v\, and similarly for the 
point b, by p2 and V2, we have' 

pi = p 2 +P2tan£ = p 2 (l+tan £); . . . . (11) 
and 

t;i+0i tan (x = V2', 
from which 

^i w (l+tan<x) n = z;2 n . ...... (12) 



164 



THERMODYNAMICS 



Multiplying equations (11) and (12), member by member, we 
obtain 

piwi n (l+tana) n = p 2 t;2 w (l+taii&); . . . (13) 

but, by construction, as stated by equation (10), 

(l+tana) n = l+tang; 

hence, equation (13) reduces to 

Vivi n =p 2 v 2 n ; 

and b is a point on the curve. In a similar manner the points 
c and i are found, etc. 




Fig. 28. 

The value of n to be used, if we are dealing with steam, is 
found by means of Zeuner's equation, which is equation (2) of 
Art. 119. And if we wish to plot an isotherm, n in equation (10) 
is made unity, thus making the angles a and g equal. 

In drawing the lines OD and OC, some convenient value for 
<x, say 15° to 20° is assumed, and the value of g is found by means 
of equation (10). 

In general, the expansion and compression curves, obtained 
by means of an indicator, for heat motors and compressors 
conform very closely to the equation 

pv n = k; 



ELEMENTARY STEAM AND ENGINE TESTS 165 

where k and n are constants for a particular process. The value 
of n, depending upon the nature of the working substance and 
the condition of operation, may lie anywhere between unity and 
1.4. 

To find the value of n from the indicator diagram we may 
proceed as follows: The pressures and volumes corresponding 
to the points a and b (Fig. 28) are determined by means of a 
scale, and then, from the equation, 

PlVi n = p 2 V2 n , 

a value for n is found. Similarly, values of n are found for a 
number of points along the curve; and these values will, in general, 
agree closely among themselves. And the mean of the values 
so found, compared with the ratio of C p to C v , for the given 
substance, is an indication of how closely the curve, under con- 
sideration, approaches an adiabatic. 

A method for finding the value of n from the indicator card, 
which has been found to give satisfactory results and is less labor- 
ious than the one just described, is as follows: Lay off the line 
OC (Fig. 28), making an angle g with the line OA; then choose 
a point on the curve, such as a, and draw the line ad parallel to 
OB, and through the point d, a line making an angle of 45° with 
the line OA, and cutting the line OC at the point h. Through 
h, now, a second line is drawn, parallel to the line OB, which 
cuts the curve at some point b; two lines, now, parallel to the 
line OA, one through a and the other through b, are drawn, and 
through g, where the line through b cuts the line OB, a line is 
drawn making an angle of 45° with the line OB and cutting 
the line, through a, at some point /. Now, this point / lies on 
a line OD, making some angle <x with the line OB. Proceeding in 
this manner, a number of points, for the line OD, are found 
which will lie very nearly in a straight line. Drawing a mean 
line through the points so found, the angle a is determined. And 
by substituting for g and a, in equation (10), n is found. 



166 THERMODYNAMICS 

It will be noticed that in this case the angle a is determined 
from the angle g and the curve under consideration; whereas, 
in the construction first given in this article the curve is deter- 
mined by means of n and the angles a and g. 

129. Behavior of Steam throughout the Cycle. When an 
engine is first started, the cylinder walls are, of course, at a 
temperature much lower than that of the steam, and conden- 
sation takes place during admission, expansion, and exhaust. 
After a time, however, permanent cyclic conditions will obtain; 
i.e., regular periodical fluctuations will have been established, 
and each cycle, so far as practical conditions permit, will 
be an exact reproduction of the cycles preceding. When these 
permanent cyclic fluctuations have been established the incoming 
steam, during admission, comes into intimate contact with the 
cylinder walls, which are at a lower temperature, due to the cooling 
of the lower pressure exhaust steam which has been in contact, 
just immediately preceding, and condensation takes place. It 
is true, that, due to wire drawing, a certain amount of drying 
takes place; but, unless the supplied steam has been super- 
heated, considerable condensation will take place during admis- 
sion, and will continue during part of the expansion-stroke; 
and may, in some rare cases, continue throughout the whole of 
the expansion-stroke. In general, however, during expansion, 
some point is reached when the temperature of the steam falls 
below that of the cylinder walls, and reevaporation takes place; 
i.e., a certain quantity of heat is abstracted, from the steam, 
by the cylinder walls, during the earlier part of the stroke, and 
a certain quantity of heat is abstracted, from the cylinder walls, 
by the steam, during the latter part of the stroke. Even if the 
quantity of heat abstracted from the cylinder walls were equal 
to the quantity of heat given up to them, which is never the 
case, there would still be a thermodynamic loss; for the heat 
abstracted from the walls is applied at a lower temperature 
than that absorbed by the walls. The heat abstracted from 



ELEMENTARY STEAM AND ENGINE TESTS 



167 



the cylinder walls, to bring about reevaporation, during exhaust, 
is completely lost, since it is all rejected to the condenser.* 

It is found, by experiment, that the exchange of heat between 
metal surfaces and perfectly dry gases, is very small even for 
considerable differences of temperature; hence, the conclusion 
that, the rapid exchange of heat between the steam and the cylinder 
walls, in a steam engine, is due to a film of conducting moisture 
which collects on the surface of the cylinder walls. 

130. Change of Dryness during Expansion. To determine 
the dryness during the expansion-stroke, it is necessary to know 
the cylinder feed and cushion steam. To determine the cylinder 
feed, the exhaust steam, for a given interval of time, is condensed 
and weighed; and for the same interval of time, the number 
of working strokes made by the engine, is determined. From 
this, the mass of steam per stroke, i.e., the cylinder feed is 
found. The cushion steam is found directly from the indicator 
diagram. 

Let, in Fig. 29, ABCD be the actual indicator diagram; OE 
and OF, respectively, the axes of zero volume and zero pressure, 




Fig. 29. 

determined as described in Art. 127, and D the point where 

the exhaust-valve has been completely closed and compression 

begins. If the assmuption be now made that, at the point D, 

the steam is saturated, no serious error is introduced; for, since 

* For a comprehensive discussion of the influence of cylinder walls, see 
" Thermodynamics of the Steam Engine," by C. H. Peabody. 



168 THERMODYNAMICS 

the mass of the cushion steam is always small in comparison 
with the total mass of steam present, during the expansion, a 
small error made in determining the cushion steam, will not 
appreciably affect the saturation curve. On the assumption 
then, that at the point D the steam is saturated, the mass of 
cushion steam is readily found, by means of steam tables; since 
its volume and pressure are given by the diagram. Taking 
the sum of the cylinder feed and cushion steam, we have the 
total mass of steam and water present during expansion; and 
from this, the saturation curve GH may be plotted. That is, 
the curve GH gives the volumes, for the various pressures, the 
steam would have occupied had it been completely saturated. 
If then, at any pressure such as 01, the horizontal line IK be 
drawn, the dryness for that pressure is at once found by the rela- 
tion 

IJ 

q= m 

131. Exchange of Heat, during Expansion, between Steam 
and Cylinder Walls. If we plot, from the p-v diagram, a T- <p 
diagram, which is easily done with the aid of steam tables, the 
transfer of heat, during expansion, between the steam and cylinder 
walls, is readily found. 

In the T-<f diagram, Fig. 30, CD is the saturation curve, 
AC is drawn at a temperature corresponding to the pressure 
at cut-off, and the point B is so located that the dryness q, for 
the point of cut-off, as found from the p-v diagram (Fig. 29), 

is given by 

AB 
q = AC 

Taking in this manner the dryness, for various pressures, on 
the p-v diagram, and transferring to the T- <p diagram, the curve 
of dryness Bnu is found. The curve ux is the curve of conden- 
sation at constant volume, and is found as described in Art. 115. 



ELEMENTARY STEAM AND ENGINE TESTS 



169 



The point n, where the vertical line pn becomes tangent to the 
curve Bnu, is the point of minimum dryness, and is given by 



<?' = 



mn 
mo' 



It is obvious from the diagram that, during the expansion, 
up to the point n, the steam is giving up heat to the cylinder 
walls; and during the remainder of the expansion-stroke, heat 
is abstracted by the steam, from the cylinder walls. Since the 
area under the curve is a measure of the heat abstracted, or 
rejected, it follows that, during the expansion, the heat given 



A B a C 


i s\ i \ 

/ / ! i \ 

/ n/ ! ^ 
m/ ii. — | 1 \o 

i ;VL ! \ 

/ ^-r'l l *> 

x<' III 

III ' 

1 ! i 

i i 

pi qi It jb 



9 
Fig. 30. 

to the cylinder walls is to that taken from them as the area 
pnBq is to the area pnut. 

As previously explained, the pressure in the cylinder during 
admission, due to wire drawing, is less than the pressure in the 
supply pipe; and as stated in Art 125, superheating may occur. 
In general, however, on account of initial priming, even if there 
were no condensation during admission, due to the incoming 
steam coming into contact with the cylinder walls of lower 
temperature, there would still be present a certain amount of 
moisture. Most authors assume, in discussing the exchange of 
heat between the steam and cylinder walls, that the steam is 



170 THERMODYNAMICS 

dry at cut-off. This assumption is neither justifiable nor nec- 
essary. No prediction can be made unless the dryness of the 
supplied steam is known. If, however, the dryness of the steam 
in the supply pipe is known, together with its pressure and the 
pressure in the cylinder, during admission, the dryness of the 
steam in the cylinder, during admission, had there been no con- 
densation, is readily computed. Let this hypothetical dryness be 
represented on the T- 9 diagram (Fig. 30) by 

Aa 

Since, however, the actual dryness at cut-off, as found from the 
indicator diagram, is 

AB 

q = AC 

it follows that an amount of condensation, represented by the 
change in entropy Ba, has taken place during admission. Hence, 
the heat given up to the cylinder walls by the steam, during admis- 
sion, is measured by the area Babq. 

Heat is also given to the cylinder walls during compression; 
this, however, is not entirely lost. Since, due to this, the tem- 
perature of the walls is raised, and the condensation during 
admission, is partially reduced. 

132. Steam Jackets. The fluctuations in temperature of the 
cylinder walls, as described in Art. 129, are the more pronounced 
the lower the speed of the engine. In other words, the higher 
the speed of the engine, the smaller the interval of time during 
which exchanges can take place between the cylinder walls 
and the steam, and as the speed becomes very high the exchange 
becomes very small. There is, however, another element to be 
considered, viz, the cooling of the cylinder, due to the fact 
that it is always at a higher temperature than the surroundings. 
This loss of heat must continually be made up by the incoming 



ELEMENTARY STEAM AND ENGINE TESTS 171 

steam; and hence, increases the condensation. This loss of heat 
is partially prevented by having the cylinder jacketed by some 
non-conducting material. In some cases, a steam-jacket is used, 
which is maintained full of live steam, • taken directly from the 
supply pipe; and therefore, the pressure of the steam, in the 
jacket is usually slightly higher than the pressure of the steam, 
during admission, in the cylinder. There is, therefore, less 
condensation, during admission, than there would be were the 
steam jacket absent; and reevaporation begins earlier. On the 
other hand, the jacket increases the area of the exposed surface; 
hence, a greater loss of heat, due to radiation. If complete 
reevaporation takes place before the exhaust-valve opens, the 
steam during the exhaust-stroke is dry, and very little heat is 
absorbed by it from the steam in the jacket. The question 
then is, whether the thermodynamic gain, obtained by applying 
the heat at a higher temperature, to bring about reevaporation 
at the earlier part of the stroke, is greater than the energy lost, 
in the jacket steam, to bring about this reevaporation, plus the 
greater radiation and heat imparted to the exhaust steam. This 
question can be answered only by experiment. Experiments 
performed, on slow and moderate-speed engines, appear to 
indicate a decided gain in economy, by using a steam-jacket. 
In a great many cases, however, such discrepant results have been 
obtained, that it is extremely difficult to say under just what 
conditions steam jackets are beneficial. 

133. Brake Power. The output of an engine of low power, 
is most conveniently measured by a friction brake, which is a 
device by means of which the power, developed by the engine, 
is absorbed in overcoming the friction applied to the surface 
of its fly-wheel; the force required to prevent rotation of the 
brake, being measured by a balance. 

The most common form assumed by the friction brake is 
depicted in Fig. 31. It consists of a number of wooden blocks 
fastened by means of bolts, to steel bands, wrapping, approx- 



172 



THERMODYNAMICS 



imately, two-thirds of the circumference of the fly-wheel. The 
wing-nut w on the bolt b makes it possible to vary the pressure 
to any desired value. The tie-rod t, going from the lower part 
of the bolt b to the lever, is merely to give rigidity to the brake. 
The rim of the fly-wheel is provided with flanges, so that water 
may be contained in it, to absorb the heat developed by the 
work done, in overcoming the friction. 

Assume, now, that the fly-wheel is rotating in the direction 
as indicated by the arrow. Then, due to friction, the brake 
will tend to rotate in the same direction; and to prevent this, 
a certain force is applied to the lever, at the point p. This force 




Fig. 31. 

is most conveniently measured by a balance; which may be 
either a spring balance or a beam balance. Let the fly-wheel be 
making N r.p.m. (rotations per minute), the net weight registered 
by the balance, to prevent rotation, be W lbs., and d be the 
horizontal distance between the center of the shaft and point 
of contact p. Then, since power is numerically equal to the 
product of angular velocity and torque, we have, employing 
the minute as the unit of time, 



P = 2%NWd ft.-lbs. per min. ; 
and since one horse-power is the equivalent of doing work at 



ELEMENTARY STEAM AND ENGINE TESTS 173 

the rate of 33,000 ft.-lbs. per minute, we have, for the brake 
horse-power, 

BJLR = ^ooo a4) 

In the case of very small units, the torque is frequently meas- 
ured by wrapping a canvas belt around the pulley, and applying 
tensions to its two free ends. The tensions are then varied, 
until the machine is loaded to the desired amount, and measured. 
The torque is then found by taking the product of the difference 
between the two tensions and the radius of pulley plus one-half 
the thickness of the belt. In this case, the heat developed by the 
work done, in overcoming the friction, is also absorbed by water 
contained in the pulley. 

When testing high-power machines, it is neither convenient 
nor desirable to make friction tests. One method used is that of 
connecting the engine under test to an electric generator, whose 
efficiency is known, and by means of an ammeter and voltmeter, 
or else by a wattmeter, determining its output. From the 
efficiency of the generator and the power delivered by it, the power 
delivered to it, by the engine, is readily found. 

Another method for determining the power delivered by an 
engine, is to make the shaft, through which the power is being 
transmitted, take the place of a transmission dynamometer. This is 
accomplished by determining the amount of twist, which a definite 
length of the shaft experiences, when transmitting the given 
power. Then, from the length and diameter of shaft, its modulus 
of rigidity, and the angle of torsion, the torque is readily found. 

134. Indicated Power. The power expended on the piston 
of an engine, by the working substance, as found by means of the 
indicator diagram, is called the indicated power. During admis- 
sion and expansion, work is being done by the working substance 
on the piston; and during exhaust and compression, work is being 
done by the piston, on the working substance. Hence, the net 
work done by the working substance, during a cycle, is measured 



174 THEEMODYNAMICS 

by the area enclosed by the indicator diagram. If then, the area 
of the indicator diagram be determined and divided by the length 
of the stroke, reduced to the proper scale, the average ordinate 
is found. The average ordinate, so found, multiplied by the scale 
of the spring, used in taking the diagram, gives the mean effective 
pressure. The area of the diagram is most conveniently found 
by means of a planimeter. There are certain types of planimeters, 
which are specially designed for determining the mean effective 
pressure from an indicator diagram. This type of planimeter is 
very convenient, inasmuch as it is only necessary to set it to the 
length of the diagram, employing a scale corresponding to the scale 
of the spring, used in taking the indicator diagram, and following 
the outline of the diagram with the tracing point of the instru- 
ment. The mean effective pressure is then given directly by 
the reading on the scale. 

The mean effective pressure is the average pressure on the 
piston, during admission and expansion, minus the average 
pressure during exhaust and compression; hence, it is the effective 
pressure, due to which external work is obtained. If the indica- 
tor spring has been calibrated to lbs. per square inch, then the 
mean effective pressure is also given in lbs. per square inch; and 
the total effective pressure on the piston is numerically equal 
to the product of the mean effective pressure and the area, 
expressed in square inches, of the piston. If we represent by P, 
the mean effective pressure, in lbs. per square inch, by A the 
area of the piston, in square inches, by L the length of the stroke 
in feet, and by N the number of cycles per minute, then the net 
work done on the piston, per minute, is 

W = PALN ft. -lbs.; 

and the indicated horse-power is 



ELEMENTARY STEAM AND ENGINE TESTS 175 

135. Mechanical Efficiency. The indicated power of an engine 
is always greater than the power delivered by the engine, by an 
amount which is equal to the power consumed in overcoming 
the engine friction. The ratio of the brake horse-power, to the indi- 
cated horse-power gives the mechanical efficiency; i.e., 

^ m = I H P ^^ 

136. Thermal Efficiency. The thermal efficiency of an engine is 
given by the ratio of the power delivered by the engine to the power 
due to the heat taken from the source. As an example, assume a 
steam engine to be taking M pounds of steam per minute from 
a boiler, the total heat of which, per pound, is H. Let the heat 
of the water in the condenser be h, which we will assume is returned 
to the boiler without losses. Then the heat, expressed in mechan- 
ical units, which is taken per minute from the boiler, is 

JM{H-h) ft. -lbs.; 

and if W represents the number of ft .-lbs. of work delivered per 
minute by the engine, then the thermal efficiency is 

W 
Eh = JM{H-h) (17) 

We will now illustrate equation (17) by a numerical example. 
Assume an engine making 300 r.p.m., doing work against a friction 
brake whose arm is 5 ft., and which requires a force of 135 lbs., 
applied at its end, to prevent rotation. If the engine is consuming 
16 pounds of saturated steam per minute, under a pressure of 80 
lbs., and returns the water without losses directly to the boiler, from 
the condenser, where the pressure is 2 lbs., what is the thermal 
efficiency? 

Substituting, in equation (17), we find 

„ 2xX300Xl35X 5 n . 

^ = 778Xl6(1182-94.2) =9 - 4 per Cent; 



176 THERMODYNAMICS 

where 1182 * is the total heat of steam under a pressure Of 80 
lbs., and 94.2 the heat of the liquid, corresponding to the temper- 
ature of the steam, under 2 lbs. pressure. 

137. Commercial Efficiency. The commercial efficiency of 
an engine is given by the ratio of the power delivered by the engine 
to the power which a perfect heat engine, working between the same 
temperature limits, would deliver. Let the symbols have the same 
significance as in Art. 136, then the work, per minute, which a 
perfect heat engine would deliver, is 

JM(H-h)^K ft.-lbs.; 

and the commercial efficiency is 

W 
Ec = „ ff (18) 

o 

Substituting in equation (18), the numerical data given as 
an illustration in the preceding article, we find 

„ 2^X300X135X5 qo 

E c = t^w = 39.0 per cent. 

778X16(1182-94.2)^1 

This is the proper method of comparison; i.e., comparing the 
actual performance of the engine with an ideally perfect engine, 
operating between the same temperature limits. 

When an engine exhausts to the atmosphere there is, of course, 
no heat returned to the boiler by means of the condensed steam, 
and the heat h, in equations (17) and (18), is lost. It is, however, 
not proper to charge this entire loss of heat against the engine; 
since, by proper arrangements part of the heat at least, contained 
by the liquid, can be returned to the boiler. 

There are other methods for rating the performance of engines, 
which are in certain cases, very convenient. One is, specifying 

* Taken from Peabody's Steam Tables. 



ELEMENTAEY STEAM AND ENGINE TESTS 



177 



the number of pounds of steam per B.H.P. hour, consumed by 
the engine. Another is, specifying the number of B.T.U. per 
B.H.P. hour, or the number of B.T.U. per K.W. hour of energy 
delivered to the bus-bar. The latter is especially expressive; 
giving, as it does, the rating of the power plant as a whole. 

138. Rankine's Cycle. Another important comparison may 
be made by the aid of Rankine's cycle, the indicator diagram of 
which is shown in Fig. 32. This indicator diagram is based on 
the assumption that the cylinder of the steam engine has no 




clearance and is perfectly insulated. AB represents the admis- 
sion at constant pressure pi, BC represents the adiabatic expan- 
sion to the pressure p2, and CD represents the exhaust, at constant 
pressure p2. 

Assume now, that we are dealing with a unit mass of liquid, 
whose specific volume is a, and that the dryness, during admission, 
is qi. If the specific volume of the steam, at the pressure pi, is 
sit "then the volume of the mixture, at the point of cut-off, is 



vi = qisi + (l- qi)a = qi(si — a) + a = qi^i+a; . 



(19) 



where \n is the increment in volume due to complete evaporation 
at the pressure pi. Since the pressure, during admission, is con- 
stant, the work done by the steam, on the piston, is 



piVi = pi(qi[Li + <s), 



(20) 



178 THERMODYNAMICS 

The work done on the piston, by the steam, during the adiabatic 
expansion, must be equal to the difference between the intrinsic 
energy of the steam before and after expansion; i.e., 

Ei-E 2 = J(hi+qm-h 2 -q2p2); .... (21) 

where Ei,hi, and pi are, respectively, the intrinsic energy, the heat 
of the liquid, and the heat of disgregation, corresponding to the 
pressure pi, and E 2 , h 2 , and g 2 are, respectively, the intrinsic 
energy, the heat of the liquid, and the heat of disgregation, cor- 
responding to the pressure p 2 . Or, to put it in another way, the 
work done on the piston, by the steam, during the expansion, is 
the difference in the heat content, expressed in mechanical units, 
before and after expansion. The work done on the piston, during 

exhaust, is 

-p 2 v 2 =-p 2 (q 2 [k 2 + c); (22) 

where q 2 and [i 2 are, respectively, the dryness and increment in 
volume, due to complete vaporization, at the pressure p 2 . Taking 
the sum of the right-hand members of equations (20), (21), and 
(22), we find, for the net work done during the cycle, 

W = J(Ap 1 qi[Li+hi-Ap 2 q 2 [L 2 -h 2 +qipi-q 2 g 2 ) + (pi-p 2 )a. (23) 

The second term of the right-hand member of equation (23) is 
very small in comparison with the other term, and may be 
neglected; hence, since from equation (55), of Art. 55, 

r=Apy.+ p, 

we find, by substituting in equation (23) the proper values, 

W = J(qir 1 +h 1 -q 2 r 2 -h 2 ) (24) 

If pi and p 2 are known, the values for hi, h 2 , n, and r 2 are found 
directly from steam tables; and by knowing qi, the value of 
q 2 is readily found by means of the T-y diagram. 



ELEMENTARY STEAM AND ENGINE TESTS 179 

From equation (24) it is readily seen that the heat converted 
into work ; is the difference between the heat taken in, during 
admission, and that rejected, during exhaust. This must neces- 
sarily follow from the assumption that there are no losses in the 
engine. But. it must be remembered that in the foregoing dis- 
cussion, the engine is considered as being independent of the boiler. 
Hence, by taking the ratio of the work actually performed by the 
engine, to the work as given by equation (24) , a result is obtained 
which serves as a basis of comparison with other engines operating 
under similar conditions. 

To illustrate, we will take the same numerical values as given 
in Art. 136, excepting that 90 per cent initial dryness will be 
assumed, instead of complete saturation. The value of r corre- 
sponding to 80 lbs. pressure is 899.8, and that corresponding to 
2 lbs. pressure is 1021.9. The dryness, after adiabatic expansion, 
at the pressure of 2 lbs. is found, by means of the T-y diagram, 
to be approximately, 75.5 per cent, hi and hi are given, respect- 
ively, by 282.2 and 94.2. Substituting these values in equation 
(24), we have 

TT = 778X16(0.9X899.8+282.2-0.755X1021.9-94.2) 

= 2,817,000 ft.-lbs. per minute. 

Taking the ratio of the work delivered by the engine, to that which 

would be realized by the Rankine cycle, we find, for the efficiency 

of the engine, 

2xX300X135X5 

2,817,000 -^Percent. 



CHAPTER XIII 
COMPOUND ENGINES 

139. If a heat engine is to convert a large fractional part of 
the heat taken from the source, into work, the temperature dif- 
ference between source and refrigerator must also be large. Hence, 
other things being equal, for a steam engine to operate econom- 
ically, it is necessary to have a large range in temperature, or what 
amounts to the same thing, a large range in pressure. 

But, when steam under a pressure of 100 lbs., and upward, is 
supplied to an engine, the fluctuations of temperature in the cylin- 
der become large; and, consequently, the condensation becomes 
excessive. To illustrate the fluctuations in temperature, assume 
an engine receiving steam under a pressure of 100 lbs., and reject- 
ing to a condenser under a pressure of 1 lb. From the steam 
curve, of Art. 101, the temperatures corresponding to these two 
pressures are, respectively, 328°F. and 102°F.; i.e., a range of 
about 226°F. Such a large range in temperature means a con- 
siderable amount of condensation during the earlier part of the 
stroke, and a consequent reevaporation during the later part 
of the stroke; but, as previously explained, this constitutes a 
thermodynamic drop; i.e., a wasteful application of heat. To 
obviate this excessive condensation, when high-pressure steam 
is used, the expansion is made to take place in two or more cylin- 
ders; and the engine is said to be a multiple-expansion, or compound 
engine. When the expansion takes place in two cylinders, the 
engine is said to be a double-expansion engine, when in three 
cylinders, a triple-expansion engine, etc. 

180 



COMPOUND ENGINES 



181 



140. Double Expansion. We will consider first the most 
simple case possible; viz, cylinders without clearance, no losses 
whatsoever, and a receiver, between the two cylinders, of such 
volume that the pressure in it, throughout the cycle, is constant. 
That is, the high-pressure cylinder receives steam from the boiler, 
which, during admission and expansion, does work on the piston. 
The steam is then rejected, at constant pressure, to the receiver; 
the pressure in cylinder and receiver, during the exhaust, being 
identical. During the same interval of time, that this is taking 
place in the high-pressure cylinder, the low-pressure cylinder 
receives an equal mass of steam, from the receiver, which in turn 
does work, during admission and expansion, on the low-pressure 
piston. The steam is then expelled, under constant pressure, 
to a condenser in which a low pressure is maintained. 

The indicator diagram, representing the foregoing is shown 
by Fig. 33. The diagrams, for the two cylinders, are drawn to the 



a B 

F ^\.E 



V 

Fig. 33. 



same scale and are superimposed. ABCD is the diagram for the 
high-pressure cylinder, and DCEF is that for the low-pressure 
cylinder. 

It will be noted that the combined diagram ABEF is precisely 
the same as would have been obtained if the expansion, from the 
initial volume, as represented by AB, to the final volume, as repre- 



182 THERMODYNAMICS 

sented by FE, had taken place in a single cylinder. That is, 
so far as indicated power is concerned, provided there are no 
losses, it is immaterial whether the expansion, from the initial 
volume to the final volume, takes place in one cylinder or a num- 
ber of cylinders. There is, however, for the case just discussed, 
due to the fact that the fluctuations in temperature have been 
reduced, a decided thermodynamic gain. There is also an impor- 
tant mechanical advantage when, other things being equal, expan- 
sion takes place in two or more cylinders. For, if the area taken 
up by the piston rod be neglected, then the stress existing in the 
rod, at any instant, is proportional to the difference in pressure on 
the two sides of the piston. This difference is a maximum, while 
admission is taking place at one end of the cylinder and exhaust 
at the other. By referring to Fig. 33, it is seen that, for the same 
ranges in pressure, the maximum difference in pressure for the 
single-expansion engine is measured by AF; whereas, for the 
double-expansion engine, the maximum differences in pressure for 
the high and low-pressure cylinders are measured, respectively, 
by AD and DF. It therefore follows, that in a compound engine, 
the piston rods may be considerably reduced in cross-sectional 
area as compared with that of a single-expansion engine. And 
since, for the same initial and final pressures the total work done 
is the same in either case, it follows that the average thrust on the 
cranks must be the same for the compound engine as it is for the 
single-expansion engine. But, in the case of the compound engine, 
the thrust is more uniformly distributed throughout the cycle; 
hence, less friction and a consequent smaller amount of wear in 
the crank bearings and joints. And, further, due to a more uni- 
form thrust, throughout the cycle, there is a smaller fluctuation 
in speed; hence, for the same uniformity of speed, the fly-wheel for 
a compound engine need not be as massive as that for a single- 
expansion engine. On the other hand, the compound engine has 
a greater number of moving parts; hence, a greater first cost, 
and additional friction. 



COMPOUND ENGINES 183 

It is obvious that the volume of the low-pressure cylinder of a 
compound engine, for the same initial and final volumes, must 
be the same as that of a single-expansion engine. The volume 
of the low-pressure cylinder is therefore fixed by the boiler pressure 
of the steam, the total expansion, the power to be developed, and 
the speed of the engine. The volume of the high-pressure cylinder, 
on the other hand, is a matter of choice; provided always that the 
ratio of the volume of the low-pressure cylinder to that of the 
high-pressure cylinder is less than the total ratio of expansion. 
The point of cut-off, however, for the high-pressure cylinder, 
as will be shown later, depends upon the ratio of the two volumes. 

From a thermodynamic standpoint, the ranges in temperature 
for the two cylinders should be about equal ; since this gives equal 
fluctuations of temperature in the cylinders. This also gives, 
very nearly, equal amounts of work done in the two cylinders; 
which, as will be seen later, is also best mechanically. Since it is 
advisable to have nearly equal ranges of temperature in the two 
cylinders, it necessarily follows that the ratio of the cylinder 
volumes is fixed by the total ratio of expansion. In practice, 
depending upon the total ratio of expansion, the ratio of the 
volume of the low-pressure cylinder to that of the high-pressure 
cylinder may vary from 3 to 5. 

'141. Tandem Compound Engine with Large Receiver. By a 
tandem compound engine, is meant an engine which has the axes 
of the two cylinders aligned; and has only one piston rod, which 
carries both pistons. In an engine of this type, the two pistons, 
necessarily, have strokes of equal lengths. Assume the volume 
of the receiver to be so large in comparison with the volume of the 
two cylinders, that there are, during the cycle, no fluctuations of 
pressure in the receiver. If, further, there be assumed no clear- 
ance and no losses whatsoever, then the indicator diagram will 
be identical with that depicted in Fig. 33. 

Let, in Fig. 34, H represent the high-pressure and L the low- 
pressure cylinder; and, further, let R be the ratio of the volume 



184 



THERMODYNAMICS 



of L to that of H. If p± be the pressure, during admission, of the 
steam in H, and p 2 the pressure of the steam in the receiver, 
which is also the back pressure on Pi, the piston of H, then, neglect- 
ing the area of the rod, the stress in the rod, between the two 
cylinders, due to these two pressures, is 



Si = (pi-p 2 )Ai; 



(1) 



where A\ is the area of Pi, and S± the stress. In a similar manner, 
the stress in the rod to the right of P2, due to the pressures p 2 
and p%, in the cylinder L, is 



S 2 = (P2-P3)A 2 ; 

P 2 L 



Fig. 34. 



(2) 



Pi H 








r 




* p? 
















« — P 2 - 




<pr 


«-*-— 















where A 2 is the area of P 2 , and S 2 the stress. Taking the sum 
of the right-hand members of equations (1) and (2), we obtain 
for the stress, in the rod r, 



S = (pi-p 2 )Ai + (p 2 -pz)A 2 . 



(3) 



Since now, the cylinders are of equal length, and R is the ratio 
of their volumes, we have 



A l =A 2 /R. 



(4) 



Substituting the value of A 1, as given by equation (4), in equation 
(3), we obtain 



S=( Vl -p 2 )~+(p 2 -p 3 )A 2 . 



(5) 



COMPOUND ENGINES 185 

Had the expansion taken place in the low-pressure cylinder 
between the pressures pi and p%, then the stress in the rod would 
have been found to be 

S' = (pi-P8)A a . ...... (6) 

Since the relation of pressure and volume of steam is not 
expressible by a simple equation, it is impossible to eliminate 
P2 from equation (5); and, therefore, no general comparison 
between S, as given by equation (5) , and S' } as given by equation 
(6), can be made. But, by assuming particular values for pi 
and p2, some idea may be obtained in regard to the relation of 
the stresses S and S'. 

To make a comparison, assume pi to be 100 lbs., and ps 1 
lb. ; and, as a matter of convenience, p2 to be 20 lbs., then p2 = Pi/5. 
Substituting this value of p2 in equation (5), we find 

*-(»-f )£+(£-»)* (7) 

Again, for the case under consideration, R will have a value 
of about 3; hence, equation (7) becomes 

4pi A 2 , piA 2 , /7pi \ A /ON 

<S= 5 X T + 5 -V* A2= \i$-Vz) A 2- • • ( 8 ) 

By comparing equations (6) and (8), it is seen that for the 
same given initial and final pressures, the maximum thrust for the 
single-expansion engine is more than double the maximum thrust 
for the double-expansion engine. 

The indicator diagram, of Fig. 33, represents an extreme case; 
and one that cannot be realized in practice. For, in the first 
place, to maintain a constant pressure, during the exhaust of the 
small cylinder and the admission to the large cylinder, requires 
a receiver of excessive bulk. Secondly, there is always a certain 
amount of resistance offered to the flow of steam in passing from 
the first cylinder to the receiver, and from the receiver to the second 



186 THERMODYNAMICS 

cylinder. Therefore, the lines representing, respectively, the 
exhaust for the small cylinder and the admission for the large 
cylinder, will not coincide. That is, the exhaust line, for the 
small cylinder, will show a higher pressure than that shown by the 
admission line for the large cylinder. This drop in pressure is, 
however, not entirely wasteful; since, due to this partially unre- 
sisted adiabatic expansion, part, and in some cases all, of the 
moisture formed in the first cylinder is removed. 

142. Compound Expansion without Receiver. In some 
engines, usually called Woolf engines, the steam passes directly 
from the one cylinder to the other. The cylinders may be either 
in tandem or side by side. It is obvious that in engines of this 
type the two pitsons must begin and end their strokes together. 
That is, the movements of the two pistons must either be in phase 
or differ by 180°. The operation is then as follows: Steam is 
admitted to the small cylinder, up to some desired fractional part 
of the stroke when cut-off takes place, and then expands to the 
end of the stroke. At the end of the stroke communication is 
established between the two cylinders, the steam begins to pass 
from the small cylinder to the large cylinder and a second expan- 
sion takes place. Since, now, the cylinders must remain in com- 
munication to the end of the stroke, there can be no cut-off in 
the low-pressure cylinder; and furthermore, since the pressures 
in the two cylinders are, at any instant, the same, it follows that 
the pressure in the small cylinder, during its exhaust, is continually 
decreasing. Admission and expansion now again take place in 
the small cylinder, while exhaust is taking place in the large 
cylinder. 

The indicator diagrams, for the cycle just discussed, if it be 
assumed that there is no clearance and no drop between the two 
cylinders, will be as represented by Fig. 35. For the high-pres- 
sure cylinder, AB represents the admission, BC the expansion, 
and CD the exhaust; and for the low-pressure cylinder, EF repre- 
sents the expansion, FG the drop during release, and GH the 



COMPOUND ENGINES 



187 



exhaust. To combine the two diagrams, the diagram EFGH must 
be drawn for a piston area corresponding to that of the diagram 
ABCD, so that equal increments in abscissas, for the two diagrams, 



A B 




i \c 


E 


k h^~- 




H 




'Lr 



' V 

Fig. 35. 



measure equal increments in volume. That part of the indicator 
diagram of the high-pressure cylinder, represented by ABC I, 
will then remain unchanged for the combined diagram. And, 
to find any point on the expansion curve, for the combined diagram 



A B 




- 1 _^G 

5. i/\ 

D ^^ 


^ ,F 







V 

Fig. 36. 

beyond the point C, corresponding to the pressure as represented 
by the point K, it is only necessary to draw a horizontal line, 
such as KN, and remember that the total volume of the steam, 
for this pressure is equal to KL+MN. This is represented in 
Fig. 36, by the line KN. In a similar manner, a number of points 



188 



THERMODYNAMICS 



are found, and the expansion curve BCF is determined. This 
gives, then, for the equivalent indicator diagram, for the two 
cylinders, the diagram ABCFGH. 

143. Tandem Compound and Small Receiver. Since it is 
impracticable to have a receiver of sufficient volume, so that the 
pressure in it is constant throughout the cycle, the cut-off, for the 
large cylinder, must be so chosen that the pressure in the receiver, 
at the end of the exhaust stroke of the small cylinder is the same 
as when release occurs, during the next stroke in this cylinder. 




Let, in Fig. 37, ABODE be the combined diagram, constructed 
as described in Art. 142; AB represents the admission to the small 
cylinder, BC the total expansion, CD the drop, during release, 
in the large cylinder, and DE the final exhaust. 

Let, now, F be the point of release for the small cylinder, then 
at the same instant admission takes place in the large cylinder; 
and, since the rate of volumetric displacement for the piston of the 
large cylinder is greater than that for the piston of the small 
cylinder, the pressure in the receiver must fall. Let FG represent 
that part of the exhaust curve, for the small cylinder, before cut-off 
takes place in the large cylinder. After cut-off has taken place 
in the large cylinder, the steam remaining in the small cylinder is 
compressed, as represented by the curve GH, into the receiver. 



COMPOUND ENGINES 189 

If, now, there is to be no drop in pressure when release occurs in 
the small cylinder, the pressures for the points F and H must be 
equal. Since H represents the pressure of the steam, when admis- 
sion begins in the large cylinder, then, by beginning with this 
point, and taking into account the total volume of the steam for 
various points, a curve such as HI, which shows the falling off 
in pressure in the large cylinder during admission, is found. And, 
where this curve, HI, cuts the curve of expansion BC, will be the 
point of cut-off for the large cylinder; the pressure at / and G 
being identical. 

As a matter of convenience, the foregoing discussion has been 
made with the assumption that the cylinders have no clearance. 
No new difficulty, however, is introduced by considering the clear- 
ance. As previously explained, a small amount of drop is not 
harmful; hence, absolute precision is not required. 

144. Cross-compound Engines. The type of double-expan- 
sion engines most frequently used, are those known as cross-com- 
pound engines. Cross-compound engines are either in twin, i.e., 
the cylinders are side by side, and the cranks make an angle of 
90° with each other, or else, the cylinders make an angle of 90° 
with each other, and the connecting rods act upon cranks in the 
same phase. In either case the piston movements are not in 
phase. That is, there is a phase difference of 90°; therefore, 
when exhaust begins to take place from the small cylinder, the 
piston of the large cylinder will not be in a position such that 
steam can be received. Hence a receiver is necessary. 

From a mechanical standpoint, the cross-compound engine 
is far superior to the tandem-compound; for, in the one case the 
cranks are actually at right angles, and in the other case, where 
the cylinders are at right angles, the mechanical effect is precisely 
the same as though the cranks made an angle of 90° with each 
other. Therefore, the turning moment, throughout the cycle, is 
much more uniformly distributed. 

The method employed to determine the point of cut-off for 



190 THERMODYNAMICS 

the large cylinder, in a cross-compound engine, so that there shall 
be no drop, is almost precisely the same as that discussed in Art. 
143. It is only necessary to take into consideration the phase 
relation of the two piston movements. 

In large steam-power plants, the cross-compound engine is 
the one most commonly employed; boiler pressures as high as 
200 lbs. to 250 lbs. being, in some cases, used. Within the last 
few years, compound engines have been operated in conjunction 
with low-pressure steam turbines; the turbine operating on the 
exhaust steam from the low-pressure cylinder. 

145. Triple Expansion. Where a more uniform turning 
moment, than that offered by a cross-compound engine, is desired, 
triple expansion is employed. In the case of triple-expansion 
engines, the cranks are frequently set so that each crank differs 
in phase by 120° from the other two; thus giving a good distribu- 
tion of turning moment. In other cases, however, the triple 
expansion takes place in four cylinders; i.e., one high-pressure 
cylinder, one intermediate cylinder, and two low-pressure cylin- 
ders. Both of these low-pressure cylinders take steam from the 
receiver, to which the intermediate cylinder rejects. The engine is 
equipped with four cranks, with a continuous phase difference 
of 90°. Triple-expansion engines have been very largely used 
in marine engineering; but, are now being superseded either by 
turbines, or else by the combination of triple-expansion engines 
and low-pressure turbines, which operate on the low-pressure 
exhaust steam from the reciprocating engines. 

From the foregoing, it is obvious that, for cross-compound 
and triple-expansion engines, it is desirable to have the work 
done in the various cylinders equal; since this gives the most 
nearly uniform turning moment for the entire cycle. And, 
as has been previously stated, the ranges of temperature, for the 
various cylinders, will be nearly equal, when the work done in the 
cylinders is equal. 

Engines having more than three stages are rare; and, it is 



COMPOUND ENGINES 191 

doubtful whether they are ever economical. Theory can merely 
serve as a guide; the final criterion being experiment. In any 
given case, however, there is a limiting value for the number of 
stages of expansion; and, in practice, this is fixed when the ther- 
modynamic gain is offset by the interest and depreciation of 
the extra capital invested, plus the extra mechanical losses 

146. Tests of Performance. The tests on compound engines 
are very similar to those on simple engines. However, when 
making tests on an engine, for efficiencies, the load should be varied 
from zero load up to, say, 25 to 50 per cent overload, and a curve 
plotted, efficiencies as ordinates and loads as abscissas. Or, 
if it be desired, the number of pounds of water per B.H.P., or 
else the number of B.T.U. per B.H.P., may be plotted against 
B.H.P. The curves may also be plotted, using I.H.P. instead 
of B.H.P. In any case, however, the curve will show the character 
of performance of the engine for the various loads; and, for what 
load the best economy is obtained. 

If the curve shows that the efficiency of the engine does not 
decrease rapidly, as the load is decreased, from the normal, then 
good service will be obtained for variable loads. On the other 
hand, if the efficiency falls off rapidly with decreasing load, then 
the engine will give good service only for approximately constant 
loads. 



CHAPTER XIV 
INTERNAL COMBUSTION ENGINES AND FUELS 

147. Whereas, in the steam engine, the combustion of the 
fuel and the application of heat to the working substance, take 
place outside of the engine, and in the internal combustion engine, 
as the name implies, combustion of the fuel, and the application 
of heat to the working substance, take place directly inside of the 
cylinder, there is, between the two types of heat motors, as regards 
the manner in which the application of heat takes place, a fun- 
damental difference. Some of the other prominent differences 
of operation between the two types of heat engines will be discussed 
later. 

The most common fuels which may be used in internal com- 
bustion engines are: Coal-oils, alcohol, natural gas, producer gas, 
blast-furnace and coke-oven gas, city illuminating gas, etc. The 
fuels most generally used are : Producer gas, natural gas, gasoline, 
petroleum and alcohol; and in all cases, there must be present 
a proper amount of air, so that sufficient oxygen is supplied, to 
bring about complete combustion. Which of these fuels is best 
depends upon a great many factors; principally upon duty, 
economy, and convenience. 

There are three typical methods for the operation of internal 
combustion engines; it being the aim to bring about the applica- 
tion of heat to the working substance, for the three different 
types, respectively, at constant volume, at constant pressure, and 
at constant temperature; thus, giving three distinct cycles, which 
will now be discussed in detail. 

192 



INTERNAL COMBUSTION ENGINES AND FUELS 193 

148. Four-phase Cycle. The four-phase * or " Otto " cycle 
was first applied to the internal combustion engine by Dr. Otto, 
in 1876; and is essentially as follows: (1) The piston is at the 
end of a return stroke with the exhaust-valve just closed and the 
inlet-valve open to a chamber where the mixing of the fuel takes 
place. The mixture consists either of gas and air, or, if a liquid 
fuel be used, of vaporized liquid and air. The piston then makes 
an outward stroke, called the aspirating stroke, and a charge of 
mixture is forced, by the external pressure, into the cylinder. 
(2) The inlet-valve is closed, a return stroke takes place, called 
the compression stroke, and the mixture is compressed until its 
volume is reduced to that of the clearance. (3) The charge is now 
ignited, usually by an electric spark, and combustion takes place 
very rapidly, and practically at constant volume, together with 
a rapid augmentation of pressure. The gaseous mixture then 
expands, forces the piston outward and does work on it. This 
stroke is called the working stroke or power stroke. (4) When 
the piston is at the end of the working stroke the exhaust-valve 
opens; and, during the return stroke, called the expulsion stroke, 
the products of combustion are expelled, and the cycle is com- 
pleted. 

It will be remembered that, in the reciprocating steam engine, 
every other stroke is a working stroke. Hence, the internal 
combustion engine, operating on a four-phase cycle, having one 
working stroke only for every four strokes, must necessarily 
during the working stroke, other things being equal, store more 
energy in the fly-wheel. The fly-wheel must, therefore, have a 
greater moment of inertia in order to carry the load properly 
during the remainder of the cycle. 

The ideal indicator diagram, for a four-phase cycle is represented 
in Fig. 38. OF is the axis of zero pressure, and AB represents 
the aspirating stroke; the pressure in the cylinder and that of 

♦Ordinarily called "four-cycle"; this, however, is not proper, since 
there are four phases to a cycle. 



194 



THERMODYNAMICS 



the atmosphere being identical. BC represents the adiabatic 
compression of the charge, and the vertical line, CD, the com- 
bustion at constant volume; OG being the axis of zero volume. 
DE represents the adiabatic expansion of the gaseous mixture 
after combustion, EB the drop in pressure to that of the atmos- 
phere, when the exhaust-valve opens, and BA represents the expul- 
sion of the products of combustion. The net work done, by the 
working substance, during the cycle, is represented by the area 
DEBC. 




O V F 

Fig. 38. 

149. Two-phase Cycle. In the two-phase cycle,* as in the four- 
phase cycle, it is the aim to bring about the application of heat, 
to the working substance, at constant volume. In the two-phase 
cycle engine, however, every other stroke is a working stroke. 

Beginning with the ignition of the mixture, there is a rapid 
rise in pressure, and then expansion of the gaseous products 
of combustion. In the meantime, a new mixture has been slightly 
compressed, either by means of an auxiliary compressor, and forced 
into a subsidiary reservoir, or else compressed in the crank case 
of the engine. Just before the piston reaches the end of the 
working stroke, an exhaust-port is opened and then, immediately 
following this, the inlet-port is opened. Since, at the opening 



* Ordinarily called "two-cycle"; which, however, is an improper 
designation. 



INTEKNAL COMBUSTION ENGINES AND FUELS 195 

of the exhaust-port, the pressure in the cylinder is slightly in excess 
of the atmospheric pressure, the gaseous products of combustion 
immediately begin to flow, out of the cylinder, through the exhaust- 
port. Immediately after the exhaust-port opens, the inlet-port, 
on the opposite side of the cylinder, is uncovered by the piston; 
and, since the mixture has been precompressed to a pressure slightly 
higher than that obtaining in the cylinder, at the end of the working 
stroke, a new charge flows into the cylinder. By a suitable arrange- 
ment the inflowing gas is directed so as to help expel the exhaust 
gases. The return stroke now begins; the piston closing first the 
inlet-port and immediately following the exhaust-port, and the 
mixture is compressed until its volume is reduced to that of the 
clearance, when ignition takes place, and the cycle is completed. 

From the standpoint of thermal efficiency, the four-phase 
cycle engine is superior to the two-phase cycle engine; but, on 
the other hand, the latter engine is far simpler in construction, 
especially in valve gearing, than the former. And, for the same 
power output, the two-phase cycle engine always has a smaller 
mass than the four-phase cycle engine. Hence, for the same power, 
the two-phase cycle engine requires less space than does the four- 
phase cycle engine. The thermal efficiency of the two-phase 
cycle engine is affected by the fact that the products of combus- 
tion are more or less imperfectly expelled; and sometimes, due 
to partial mixing in the cylinder of the new charge and the products 
of combustion, the exhaust gases may contain unburnt fuel. 
Furthermore, it may happen that complete combustion has not 
taken place when the inlet-port is opened, thus causing premature 
ignition. 

150. The Brayton Cycle. In the Br ay ton cycle, it is aimed to 
bring about the application of heat at constant pressure. In 
its operation the Brayton engine compresses air in a separate 
cylinder and stores it in a receiver. This compressed air is admit- 
ted through a mass of felt, charged with crude petroleum, to the 
cylinder of the engine. An auxiliary valve is continuously open, 



196 THERMODYNAMICS 

permitting a very small jet of air, charged with petroleum, to 
flow into a small chamber communicating with the cylinder, 
where it burns continuously while the engine is in operation. 
When the main inlet-valve opens, air rushes through the felt, 
takes up a charge of petroleum, and is ignited by the small flame. 
Since, now, there is direct communication between the cylinder 
and the air reservoir, the pressure in the cylinder cannot rise above 
that in the reservoir; and therefore, combustion takes place at 
practically constant pressure. The fuel supply is then cut off, 
and expansion takes place, approximately adiabatically, until 
the pressure has fallen, depending upon valve adjustment, to 
any desired value. The exhaust-valve is then opened, and 
during the return stroke the products of combustion are expelled; 
hence, this cycle is a two-phase cycle. The Brayton engine has 
practically been superseded by the one which will be discussed 
in the next article. 

151. The Diesel Cycle. In the Diesel engine the aim is to 
bring about combustion, and therefore the application of heat, 
at constant temperature. The Diesel cycle is as follows : During 
the aspirating stroke a charge of air flows into the cylinder, where 
it is compressed, approximately adiabatically, to a very high 
pressure, during the compression stroke. The inlet-valve now 
opens and the fuel in the form of oil, usually crude oil, is injected 
and immediately becomes ignited, due to the high temperature 
of the air, caused by the high precompression. Combustion now 
takes place at nearly constant temperature, the piston advances, 
and the expansion of the gases, up to the point when the fuel 
is cut off, is nearly isothermal. From this point, up to where the 
exhaust-valve opens, the expansion is approximately adiabatic. 
The exhaust-valve being fully open, the pressure falls to that of 
the atmosphere, the expulsion stroke takes place, and the cycle 
is completed. The Diesel cycle, like the Otto cycle, is a four- 
phase cycle. 

The ideal indicator diagram, of the Diesel engine is shown in 



INTEENAL COMBUSTION ENGINES AND FUELS 197 

Fig. 39, in which A B represents the aspirating stroke, BC the 
adiabatic compression, CD the isothermal expansion, DE the 
adiabatic expansion, EB the drop in pressure when the exhaust- 
valve opens, and BA the expulsion stroke. Theoretically, the 
adiabatic expansion DE may be carried up to a point when the 
pressure of the gases has fallen to that of the atmosphere. But, 
for the same reason as was given in discussing the steam engine, 
in Art. 115, this is not economical. 

Theoretically, the Diesel cycle is the most efficient cycle, so far 
used, in the operation of internal combustion engines; realizing, 




v 
Fig. 39. 



as it does, the application of heat at a practically constant temper- 
ature. Thus approaching, as regards the absorption of heat, the 
Carnot cycle. The high precompression, however, requires that 
the engine have a very massive fly-wheel, and subjects it to severe 
strains. Furthermore, to start the engine, a reservoir of com- 
pressed air is required, operating on it very much as steam operates 
on a steam engine; and when the engine is up to speed, the valves 
are shifted, so as to disconnect the air, and admit the oil. How- 
ever, by means of Diesel motors of 300 H.P. capacity, thermal 
efficiencies as high as 32 to 33 per cent have been obtained. 



198 THERMODYNAMICS 



Fuels and Fuel Tests 



152. Before discussing, mathematically, the ideal indicator dia- 
gram of the internal combustion engine, it will prove instructive 
to consider briefly the chemical behavior of the fuel as combus- 
tion takes place ; and also to compare the volume of the mixture, 
before combustion, with the volume of the products of com- 
bustion. For the volumetric comparison, the same temperature 
and pressure must, of course, be assumed before and after combus- 
tion; and we must know what volume the fuel, in the vaporized or 
gaseous condition, will occupy. Furthermore, from the chemical 
constitution of the fuel, we must determine the quantity of oxygen 
which has to be supplied to bring about complete combustion, 
and the volumetric changes due to changes in chemical con- 
stitution. 

153. Chemical Reactions. Experiment shows that, when two 
or more gases react chemically to form a gas or gases of different 
chemical constitution, the numbers, representing the volumes of 
the combining gases, are fixed with respect to each other by definite 
simple ratios; and, likewise, the volume or volumes obtained, 
after the chemical reaction has taken place, are definitely fixed by 
the volumes of the combining gases. The foregoing is best illus- 
trated by the consideration of a few concrete cases. As an 
example, when the gases hydrogen and oxygen react chemically 
to form steam, then for every given volume of oxygen there is 
required double the volume of hydrogen; or, to put it numer- 
ically, two liters of hydrogen combine with one liter of oxygen 
to form two liters of dry steam. In the notation adopted by 
chemists, this chemical action is expressed as follows : 

H 2 +0 = H 2 (1) 

The volumetric relations are, of course, only true for identical 
temperatures and pressures. Likewise, one liter of carbon vapor 



INTERNAL COMBUSTION ENGINES AND FUELS 199 

unites with an equal volume of oxygen to form two liters of 
carbon-monoxide; and the two liters of carbon-monoxide will 
combine with one liter of oxygen to form two liters of carbon- 
dioxide. The reaction is expressed symbolically by 

CO+0 = C0 2 (2) 

Finally, as another example, six volumes of carbon vapor unite 
with six volumes of hydrogen to form two volumes of benzene; 
which is given by 

C6+H 6 = C 6 H6 (3) 

It will be noticed that in all cases, excepting that of two ele- 
mentary gases of equal volumes, there is, when the gases unite 
chemically, a reduction in volume; but, in every case, and this 
is generally so, the volume of the combined gases is two units 
of the measure of volume chosen. It must, however, not be under- 
stood, from the foregoing equations, that in all cases the compound 
is formed in as simple a manner as here expressed; for, interme- 
diate steps are frequently necessary. 

It is of interest to note that, if the density of one of the gases 
be chosen as unity, and the relative densities of the various gases 
be known, the density of the compound gas may immediately 
be found from its chemical formula.* Thus, if the density of 
hydrogen be taken as unity, which is the most convenient, its 
density being less than that of any other gas, then the density 
of oxygen, for the same temperature and pressure, is approximately 
16. Hence, we find, from equation (1), since three volumes, 
having a combined mass of 18, reduce to two volumes, the density 
of dry steam, with respect to hydrogen equals 9. In a similar 
manner, from equation (2), since carbon vapor has 12 times the 

* The determination of the chemical constitution of compounds is 
usually attended by extreme difficulties; but it is here neither possible nor 
is it essential to describe the various methods used. For complete descrip- 
tions and discussions, the student is referred to standard works on physical 
chemistry. 



200 THERMODYNAMICS 

density of hydrogen, the density of carbon-dioxide is found to 
be 22; and, from equation (3), the density of benzene is found to 
be 39. 

154. Gasoline. The most volatile of all the fuel oils obtained 
from petroleum, by fractional distillation, is gasoline. The com- 
position of gasoline is somewhat variable; but, its chemical con- 
stitution is represented with sufficient accuracy by the formula 
CeHi4, and its density by 0.7. Assuming the formula CeHi4, 
then the chemical equation, representing complete combustion, 
is given by 

C 6 Hi4+190 = 6C02+7H 2 (4) 

Since the temperature in the cylinder of an internal com- 
bustion engine is of sufficient intensity to insure complete vapor- 
ization of the fuel, the gasoline vapor will behave as a gas; and 
since two volumes of gasoline vapor will yield, upon decomposi- 
tion, six volumes of carbon and fourteen volumes of hydrogen, 
nineteen volumes of oxygen, under the same condition, as regards 
temperature and pressure, must be supplied to bring about com- 
plete combustion. Twelve of these volumes of oxygen will com- 
bine with the carbon, to form twelve volumes of carbon-dioxide, 
and seven volumes will combine with the hydrogen to form four- 
teen volumes of dry steam. Since the average composition of 
the air, by volume, is 21 per cent oxygen and 79 per cent nitrogen, 
it follows that for every volume of oxygen supplied there will be 
present 79/21 volumes of nitrogen. Hence, assuming that the 
mixture, before combustion, contains two volumes of gasoline 
vapor, then its total volume is 

79 
2+19+— X19 = 92.5 volumes (very nearly). 

And after combustion the volume, referred to the same temper- 
ature and pressure, is 

79 
12+14+— X 19 = 97.5 volumes (very nearly). 



INTEKNAL COMBUSTION ENGINES AND FUELS 201 

Hence, the volume of the products of combustion, referred to the 
same temperature and pressure, if just enough air be present to 
bring about complete combustion, is approximately 5.4 per cent 
in excess of the volume of the mixture. Experience, however, 
shows that, for an internal combustion engine to operate satis- 
factorily, the quantity of air supplied to it must be considerably 
in excess of that required for complete combustion. The excess 
may vary from 15 to 50 per cent; and this, together with the neu- 
tral gases, remaining in the cylinder after the expulsion stroke, 
reduces the difference between the volumes, before and after 
combustion, appreciably. 

155. Kerosene. The most important of the fuel oils, obtained 
by the fractional distillation of petroleum, and representing about 
50 per cent of the total yield, is kerosene. Kerosene is consider- 
ably less volatile than is gasoline, has an average density of about 
0.805; and its chemical constitution is represented, with a fair 
degree of accuracy, by the formula C10H22. The chemical equa- 
tion, representing complete combustion, then is 

CioH 2 2+310 = 10C02+llH 2 0; . ... (5) 

and by the method used, when dealing with gasoline, we find for 
the volume of the mixture, before combustion, 

79 
2+31+^rX31 = 149.6 volumes (very nearly). 

After combustion, the volume, referred to the same temperature 
and pressure, is 

79 

20+22+^X31 = 158.6 volumes (very nearly). 

Hence, if the quantity of air present carries just a sufficient amount 
of oxygen to bring about complete combustion, then the volume 
of the products of combustion is 6 per cent in excess of the volume 



202 THERMODYNAMICS 

of the mixture. This percentage difference is, of course, reduced 
somewhat by an excess of air and neutral gases being present 
during the cycle. 

That the volumes of the gases, before and after combustion, 
are found to be nearly equal is due to the fact that the volume 
of inert nitrogen, which is necessarily present, is always large in 
comparison with the volume of the other gases. In the two cases 
just discussed, viz, gasoline and kerosene, the volume of the gases 
after combustion was found to be greater than the volume of the 
mixture. This, however, is not necessarily the case; for, if 
we consider carbon-monoxide as a fuel we shall find conditions 
reversed. The chemical equation, expressing complete com- 
bustion for CO, is 

CO+0 = C0 2 ; (6) 

from which we find, respectively, for the volumes of the mixture and 
products of combustion, approximately 6.8 and 5.8. This gives 
the volume, after combustion, approximately 14.7 per cent less 
than the volume of the mixture. CO is a gas of much lower 
density than is either gasoline or kerosene vapor; and, by consider- 
ing the two following cases, it becomes manifest that, other things 
being equal, the higher the density of the fuel gas, the greater the 
volume of the gases, after combustion, in comparison with the 
volume of the mixture. Assume the two fuels to be, respectively, 
C2H2 and CeH6. For the former we have 

7Q 7Q 

C 2 H 2 +50+^X5N = 2C0 2 +H 2 0+^X5N; . . (7) 

and, for the latter, 

7Q 7°, 

C 6 H 6 +150+^X15N = 6C0 2 +3H 2 0+^X15N. . (8) 

From ^equation (7), we find, for the volumes, before and after com- 
bustion, respectively, 25.8 and 24.8; and, from equation (8) we 
find, for the volumes, before and after combustion, respectively, 



INTERNAL COMBUSTION ENGINES AND FUELS 203 

73.4 and 74.4. That is, in the latter case, where the density of 
the fuel is greater, the volume of the gases after combustion, in 
comparison with the volume of the mixture, is greater than it 
is in the former, for the less dense fuel. In general, the fuel con- 
sists principally of carbon and hydrogen; and the products of 
combustion consist of CO2 and H2O vapors. Hence, if the fuel 
is comparatively rich in hydrogen, then, since the H2O vapor 
is of a much lower density than is CO2 vapor, it follows that the 
products of combustion occupy a greater volume than they would, 
were they produced, from a fuel poor in hydrogen. 

156. City Gas. The gases, used for illuminating purposes, 
in different cities, vary considerably in composition. Not only 
is there a variation in going from one plant to another, but the gas 
drawn from the supply main will be found to vary somewhat for 
different parts of the day. The following table is a fair average 
for the composition, by volume, of the gas supplied in the Borough 
of Manhattan : 

Per Cent. 

Carbon-dioxide (C0 2 ) 1.9 

Illuminants (practically C2H4) 9.9 

Carbon-monoxide (CO) 18.2 

Methane (CH 4 ) 22.75 

Hydrogen (H 2 ) 42. 

Nitrogen (N 2 ) 5.25 

It is, therefore, impossible to make computations, of any value, 
with respect to city gas, unless samples of the gas supplied to the 
engine, are subjected to chemical analysis. 

157. Calorific Value of Fuel. In general, to decompose a 
compound, into its constituents, requires an expenditure of energy 
in the form of heat; hence, unless the fuel is in a form such that 
it can combine directly with oxygen, without first being decom- 
posed, the available heat, i.e., the calorific value of the fuel, is less 
than that developed by its combination with oxygen. Further- 



204 THERMODYNAMICS 

more, if the fuel is in the liquid form, heat is absorbed to convert 
it into a vapor. In general, however, the heat of vaporization is 
negligibly small in comparison with the other quantities involved. 
As a matter of illustration, we will assume that methane (CH4) 
is used as a fuel. The chemical equation, for complete com- 
bustion, is 

CH 4 +40 = C0 2 +2H 2 0. ....... (9) 

In Art. 29, it was stated that when 1 gram of H combines with 
to form H2O, about 34,000 gram calories are evolved, and dur- 
ing the combination of 1 gram of C with O to form CO2, about 
8000 gram calories are evolved. Assuming now, as a matter of 
convenience, 1 gram of CH4, then since the density of carbon 
vapor with respect to hydrogen is 12, we will have 0.25 grams of 
H and 0.75 grams of C. Therefore, the complete combustion 
of 1 gram of CH4, if there were no heat required to decompose 
the compound, would yield 

0.25X34,000+0.75X8000 = 14,500 gram calories. 

Experiment, however, shows that, when 1 gram of CH4 is con- 
sumed, to form H2O and CO2, approximately 13,200 gram calories 
are evolved. Hence, the difference, viz, 1300 gram calories are 
absorbed in decomposing the compound into its elements. 

It must always be remembered that, the experiments conducted 
for the purpose of determining heats of combustion and decom- 
position of a fuel, though simple in operation, may be attended 
by difficulties of a chemical nature; and therefore, in general, 
the values found in tables, must be taken as being only approx- 
imations. 

158. Determination of Calorific Values. The determination 
of the calorific value of a gaseous fuel is a very simple experiment, 
provided there is available a modern gas calorimeter. The gas, 
from the source of supply, is first passed through an accurately 



INTERNAL COMBUSTION ENGINES AND FUELS 205 

calibrated meter, and from this through a pressure regulator which 
maintains, throughout the progress of the experiment, the pressure 
of the gas constant. From the pressure regulator, the gas is 
supplied to a Bunsen burner where combustion takes place. 
The air has free access to the chamber in which the burner is 
placed, such that a sufficient quantity of oxygen is supplied to 
insure complete combustion. From the combustion chamber, the 
products of combustion pass through a system of tubing, sur- 
rounded by circulating water, in a manner such that the tempera- 
ture of the exhaust gases is reduced to that of the outflowing 
water, which is practically the same as that of the surroundings, 
when exhaust takes place. The inflowing water, by means of 
proper arrangements, is maintained at practically a constant pres- 
sure and temperature; and hence, when steady conditions have 
been attained, the temperature of the outflowing water will also 
be practically constant. 

The exhaust gases will consist mainly of CO2; since the H2O, 
which is formed by combustion, is condensed. After steady 
conditions have been assumed, i.e., the water due to condensa- 
tion is flowing at a constant rate, and the temperatures of the water 
at inflow and outflow are sensibly constant, the outflowing cir- 
culating water is caused to flow into a receptacle of known weight, 
and at the same time the water of condensation is caused to flow 
into a second receptacle of known weight. Simultaneously with 
these two operations, a reading is taken on the gas meter and the 
temperatures of the circulating water at inflow and outflow are 
noted. The temperatures of the water at inflow and outflow 
are then noted at suitable intervals of time, until a sufficient 
quantity of circulating water has been collected. A second 
reading is then taken on the gas meter, and, simultaneously with 
this, the collection of water in the two receptacles is discontinued. 

From the initial and final readings of the gas meter, the volume 
of the gas consumed, corresponding to the existing pressure and 
temperature, is found; and this volume, corrected to a tempera- 



206 THERMODYNAMICS 

ture of 62°F.* and a pressure equal to 30 inches of mercury, 
gives the volume, as regards American gas engine practice, for 
standard conditions. From the mean of the temperatures for 
inflow and outflow, together with the quantity of circulating 
water collected, the quantity of heat, absorbed by the circulating 
water, is readily found. 

Let during the progress of the experiment, ti and T2, respect- 
ively, be the mean of the temperatures of inflow and outflow, and 
M the mass of circulating water collected, then the quantity of 
heat absorbed, by the circulating water, is 

Q' = J|f( T2 -Ti) (10) 

Q', in equation (10), represents the heat of combustion, plus the 
heat of condensation. Since, however, the temperature of the 
exhaust gases, in the case of any internal combustion engine, 
is always higher than that of condensation, corresponding to 
the existing pressure, of the H2O vapor present, it follows that 
the heat available for doing work in the engine is less, for the 
given quantity of gas consumed, than is indicated by equation 
(10). To obtain the available heat, there must be deducted from 
the heat absorbed by the circulating water, the heat due to con- 
densation, plus the heat given up by the condensed steam in 
cooling from the temperature of condensation to the temperature 
of the outflowing water. Let m be the mass of steam condensed, 
t the temperature at which condensation takes place, and r the 
corresponding heat vaporization. The quantity of heat given 
up to the circulating water, by the steam and condensed water, 
then is 

q = m\r+(z— T2) ! (11) 

* There is no gain by using 62 °F. in place of 0°C. For, the temperature 
is seldom 62 °F.; hence, if accurate results are required, a correction for 
temperature is necessary. It therefore would be more convenient if 0°C. 
were at all times used, since this is the temperature used by physicists and 
chemists, as the standard temperature, for which the volumes of gases are 
specified. 



INTERNAL COMBUSTION ENGINES AND FUELS 207 

Subtracting equation (11) from equation (10), we find the avail- 
able heat, 

Q = ilf(T 2 -Ti)-mir+(T-T 2 )! (12) 

Dividing by V, the volume of gas consumed, we find the available 
heat per unit volume; i.e., the calorific value, is 



^ M(T2-Ti)-m{r+(T-T 2 )i 



The calorific value is usually specified in B.T.U. per standard 
cubic foot of gas. 

The temperature of condensation is approximately the same 
as the normal boiling-point, since, the pressure in the calorimeter 
does not vary greatly from that of the atmosphere. However, 
since q is always small in comparison with Q r , no serious error 
can be introduced in the final result by not estimating the value 
of t with absolute precision. 

159. Liquid Fuels. In the case of liquid fuels, the calorific 
value may be determined in precisely the same manner as that of 
gases. It is only necessary to have an accessory piece of appar- 
atus, by means of which the liquid is converted into a gas or vapor; 
the quantity of fuel consumed being determined by weighing. 
In the case of liquid fuels, however, the calorific value is usually 
specified in B.T.Us. per pound of fuel. 

As a matter of interest, a table is here appended, in which 
is given the lower calorific values of various substances; i.e., that 
calorific value is given for each substance which would have to 
be taken if it were considered as a fuel for an internal combustion 
engine. The calorific values, given in this table, are approx- 
imately correct for the first three figures; and the chemical 
formulae for gasoline and kerosene, as previously stated, are only 
approximations. For gaseous mixtures, such as city gas, no 
figures for the calorific value can be given on account of the vari- 



208 



THERMODYNAMICS 



ablity in constitution. Actual experiment, by means of a Junker's 
calorimeter, shows the calorific value of illuminating gas as fur- 
nished to the Borough of Manhattan, to be about 590 B.T.U. 
per cubic foot. 

CALORIFIC VALUES 





Chemical 
Formula. 


Gram 
Calories, 
Per Gram. 


jBritish Thermal Units. 


Gas. 


Per Pound. 


Per Cu. Ft. 




at 32° F. 


at 62° F. 


Hydrogen 


(H 2 ) 

(CH 4 ) 
(C2H2) 
(C2H4) 
(C 2 H 6 ) 
(C 4 H 8 ) 
(C C H 6 ) 

(CO) 

(C 2 ) 

(C 6 H 14 ) 
(C 10 H 2 2) 
(C 2 H 6 0) 
(CH4O) 


28900 

11900 

11700 

11400 

11300 

10800 

9610 

2430 

8110 

10300 

10100 

6560 

4750 


52000 

21400 
21100 
20500 
20300 
19400 
17300 
4380 
14600 

18500 

18200 

11800 

8550 


291 
957 
1530 
1600 
1700 
3030 
3770 
343 


273 


Methane 


898 


Acetylene 


1440 


Ethylene 


1510 


Ethane 


1600 


Butylene 


2850 


Benzene 


3540 


Carbon-monoxide . . 
Carbon 


322 


Liquid 
Gasoline 




Kerosene 

Ethyl Alcohol 

Methyl Alcohol .... 





THERMAL CAPACITIES AND DENSITIES OF GASES 



Chemical 
Formula. 



Grams per 

c.c. for 

Standard 

Pressure 

at C C. 



I Thermal Capacities, 


Gram Calories Per Gram. 


L p 


C v 


0.2375 


0.169 


0.217 


0.154 


0.244 


0.173 


3.409 


2.42 


0.242 


0.173 


0.217 


0.168 


0.593 


0.449 


0.429 


0.346 


0.530 


0.402 


0.480 


0.361 



Ratio. 



kp/^t 



Air 

Oxygen 

Nitrogen 

Hydrogen 

Carbon-monoxide . 
Carbon-dioxide . . . 

Methane 

Ethylene 

Ammonia 

Superheated steam 



(0 2 ) 

(N 2 ) 

(H 2 ) 

(CO) 

(C0 2 ) 

(CH 4 ) 

(C2H4) 

(NH 3 ) 

(H,0) 



0.001293 

0.001429 

0.001255 

0.00008955 

0.001250 

0.001965 

0.000715 

0.001251 

0.0007616 



1.405 

1.41 

1.41 

1.41 

1.40 

1.29 

1.32 

1.24 

1.32 

1.33 



INTERNAL COMBUSTION ENGINES AND FUELS 209 

The values for thermal capacities given in the foregoing table 
are the results obtained from experiments conducted, in general, 
between the limits of 0°C. and 200°C. In some cases the ranges 
were very small; hence, the given values are not necessarily true 
for high temperatures. 

The various values given in the two foregoing tables have 
mostly been taken from Landolt and Bbrnstein's tables. 



CHAPTER XV 

IDEAL COEFFICIENT OF CONVERSION AND ELEMENTARY 

TESTS 

160. In Art. 102 it was shown what fractional part of the heat 
abstracted from the source, can be converted into work by a perfect 
steam engine and boiler, and in Art. 138, a perfect engine was 
considered independently of the boiler. Either of these results 
may be employed as a standard, depending upon whether we 
are considering the engine and boiler jointly or the engine alone. 
Similarly, some ideal standard of performance, by means of which 
internal combustion engines may be compared, must be assumed. 
The ideal indicator diagram based upon the four-phase cycle, has 
been found convenient; since this cycle is the one most generally 
used. 

161. Ideal Indicator Diagram. To deduce an expression 
for the maximal amount of work that can be realized during a 
cycle, by means of an internal combustion engine, it is, of course, 
necessary to assume perfect conditions. The following assump- 
tions will be made: The compression of the mixture is adiabatic, 
the combustion of the fuel, and therefore the application of heat, 
takes place at constant volume, the expansion of the products 
of combustion is adiabatic, the rejection of heat takes place at 
constant volume, and the thermal capacity of the gas is constant 
throughout the process. A further assumption, which by a 
previous discussion, under the heading of fuels, has been shown 
to be approximately true, will have to be made; viz, that for the 
same temperature and pressure, the volume occupied by the 

210 



IDEAL COEFFICIENT OF CONVERSION AND TESTS 211 

gaseous mixture, before combustion, is equal to the volume of 
the products of combustion. 

In Fig. 40, 01 and OH are, respectively, the axes of zero pres- 
sure and zero volume, and the various parts of the cycle are repre- 
sented as follows: AB is the aspirating stroke at constant pressure, 
BC the adiabatic compression of the mixture, CD the combustion 
and application of heat at constant volume, DE the adiabatic 
expansion of the products of combustion, EB the rejection of 
heat at constant volume, and BA the expulsion stroke at con- 
stant pressure. Since, during the aspirating and expulsion strokes, 



H 



D 


f\ 2 




C 


Ti X^ 






^^^Se 


A 




-^-— — _t 4 




"* B 



V I 

Fig. 40. 



the pressures on the two sides of the piston are assumed equal, 
the net work done, during the cycle, is obviously measured by the 
area DEBC. If, now, we represent by T 1} T 2 , T 3 , and T±, the 
temperature of the working substance for the points C, D, E, and 
B, the heat applied during combustion, while the pressure rises 
from that represented by the point C to that represented by the 
point D, is 



Qi=MC,(T2-T{); 



CD 



where M is the mass of the mixture, C v the thermal capacity per 
unit mass at constant volume, and Qi the heat developed during 
combustion. The heat rejected at constant volume, while the 



212 THERMODYNAMICS 

pressure falls from that represented by the point E, to that repre- 
sented by the point B, is 

Q 2 =MC v (T z -n); (2) 

where Q 2 is the heat rejected by the working substance, while 
the temperature changes from T3 to T4. It is immaterial whether 
the change in temperature, from T% to T4, since the final result 
is precisely the same, takes place inside or outside of the cylinder. 
The conditions are analogous to those discussed, in Art. 114, for 
condensation at constant volume in the steam cylinder. 

Since, now, there is no exchange of heat during the adiabatic 
expansion DE, and, likewise, during the adiabatic compression 
BC, and the thermal capacity of the mixture is approximately 
equal to that of the products of combustion, it follows, from 
equations (1) and (2), that the heat converted into work, during 

the cycle, is 

Q 1 -Q 2 =MCUT 2 -T 1 )-(T 3 -T^)}. ... (3) 

And, since the ratio of the heat converted into work to that 
abstracted from the source, is the ideal coefficient of conversion, 

we find 

= Qi-Q2 = MC v \(T2-T 1 )-(Ts-T^)\ 
*> Qi MC V (T2-T!) 

from which 

^-JWY ••••••• (4) 

Since both DE and BC represent adiabatic changes for the same 
changes in volume, we have, from equation (50), Art. 49, 
' V2 y-i Tz T± T3-T4 



>\ n - 1 = T 1= T4 L = 
7 T 2 Tx 



vil T 2 T x T 2 -T^ 

where v\ is the volume of the mixture before compression, and 
v 2 the volume after compression. Hence equation (4) becomes 

,-i-^r'. in, 



-©■• 



IDEAL COEFFICIENT OF CONVERSION AND TESTS 213 

Equation (5) shows that the ideal coefficient of conversion is 
a function of the ratio of the volume before compression to the 
volume after compression; and increases with the amount of 
precompression. 

162. Theoretical Temperatures. The temperature which 
would obtain upon complete combustion, if there were no losses, 
is readily computed for any given case. That is, the theoretical 
rise in temperature, viz, T2 — T1, is numerically equal to the ratio 
of the heat of combustion to the thermal capacity of the products 
of combustion. However, it is found to be necessary, in order 
to have proper lubrication between the piston and cylinder 
walls, so as to prevent deterioration of material, to abstract heat 
from the cylinder walls, either by water jacketing, or else by air 
cooling. The former, that is water cooling, is brought about by 
having water, at a comparatively low temperature, circulate in a 
jacket surrounding the cylinder; and the latter, viz, air cooling, 
is brought about by increasing the surface of the exposed part of 
the cylinder by means of ribs, and having a stream of air playing 
over it continuously, by means of an air blower of some kind, or 
else, as is the case in some automobile engines, the circulation of 
air is brought about by the motion of the car. In any case, the 
heat abstracted, due to either water or air cooling, limits the rise in 
temperature. Therefore, the temperature found in the cylinder 
of an internal combustion engine, is always less than that pre- 
dicted from the heat of combustion and the thermal capacity of 
the products of combustion. Frequently, the actual temperature 
is found to be only 50 per cent of the theoretical temperature. 

163. Standard Diagram. In deducing the expression for the 
ideal coefficient of conversion for the internal combustion engine, 
Art. 161, certain assumptions, in regard to thermal capacities 
and volumes before and after combustion, were made, which are 
only approximations. But the errors involved in these assump- 
tions are very small in comparison with the difference between 
the actual and theoretical temperatures obtaining in the cylinder. 



214 THERMODYNAMICS 

However, the diagram described in Art. 161, and the results 
deduced therefrom, though differing materially from what can 
be realized in practice, are very convenient as a basis for com- 
paring the performances of internal combustion engines. 

Elementary Engine Tests 

164. Brake Power and Indicated Power. The power delivered 
by an internal combustion engine is determined in precisely the 
same manner as is that of a steam engine. This has been fully 
described in Art. 133. Furthermore, the indicated power of an 
internal combustion engine is also found in the same manner as 
is that of a steam engine, as described in Art. 134. But it must 
be emphasized that N, in equation (15) of Art. 134, represents 
not the number of revolutions per minute of the fly-wheel, but 
the number of cycles per minute in the cylinder under test. 

The ratio of Brake Power to Indicated Power is, of course, 
in the case of an internal combustion engine, as well as in the case 
of a steam engine, a measure of the mechanical efficiency. It is 
found, however, that the mechanical efficiency of an internal 
combustion engine, other things being equal, is always less than 
the mechanical efficiency of a steam engine. This is principally 
due to the fact that, owing to the high temperatures existing in 
the cylinders of internal combustion engines, the lubrication is 
not as good as that obtained in steam cylinders. 

165. Thermal Efficiency of Internal Combustion Engine. 
The thermal efficiency of an internal combustion engine is, of course, 
the ratio of the power delivered by the engine, to the power due 
to the fuel consumed. In making a test, the engine is loaded by 
means of a brake, or some other contrivance, to the desired 
amount. Then, in the case of a gaseous fuel, the volume of gas 
consumed is measured by means of a meter. Simultaneously with 
this, as described in Art. 158, the calorific value of the gas is 
determined. The best results are obtained if continuous tests 



IDEAL COEFFICIENT OF CONVERSION AND TESTS 215 

are made for the calorific value of the fuel; that is, if the 
supply to the fuel calorimeter is tapped directly onto the main, 
supplying fuel to the engine, and samples of the fuel are tested, 
for calorific values, throughout the entire run. The ratio, then, 
of the work done by the engine, during the test, to the work, 
expressed in the same units, due to the fuel consumed, which is 
equal to the product of the volume of gas consumed during the 
run and the mean calorific value of the gas, as found by means of 
the gas calorimeter, is a measure of the thermal efficiency. Or, 
if a liquid fuel be used, the work due to the fuel consumed, is found 
from the product of the mass of liquid consumed, during the run, 
and the mean calorific value per unit mass. The calorific value, 
per unit mass of the liquid, is determined as described in Art. 159. 
166. Actual Indicator Diagram of Internal Combustion Engine. 
By means of the indicator diagram, taken from an internal com- 
bustion engine, the behavior of the working substance may be 
conveniently studied. The actual indicator diagram differs, 
of course, from the ideal indicator diagram, as depicted in Fig. 40. 
Whereas, in the ideal indicator diagram, the line representing 
the aspirating stroke, is parallel to the axis of zero pressure, in the 
actual indicator diagram the line representing the aspirating 
stroke approaches the axis of zero pressure, as represented by 
AB in Fig. 41. This is due to the throttling effect of the inlet- 
valve, on account of which, the pressure in the cylinder decreases 
as the piston advances. In Fig. 41, 01 and OH are, respectively, 
the axes of zero pressure and of zero volume, and A A' is the atmos- 
pheric line. The curve AB, as just stated, represents the aspira- 
ting stroke; and shows the pressure in the cylinder at the end 
of this stroke, less than the atmospheric pressure, by an amount 
A'B. The compression of the mixture, which is approximately 
adiabatic, is represented by the curve BC. The combustion of 
the mixture, and consequent rise of pressure in the cylinder, is 
represented by the curve CD, which is, if the ignition has been 
properly timed, practically parallel to the axis OH. DE is the 



216 



THEEMODYNAMICS 



curve representing the expansion of the products of combustion. 
At E the exhaust-valve begins to open, and the pressure decreases 
rapidly to the end of the stroke F. The expulsion stroke then 
begins, and the pressure continues to decrease rapidly up to the 
point G. At this point, the exhaust-valve is fully open and the 
pressure decreases gradually as represented by the curve GA, to 
the end of the stroke, where the pressure is practically atmos- 
pheric, and the cycle has been completed. The pressure in the 
cylinder, during expulsion, is higher than that of the atmosphere 
due to the resistance offered by the exhaust-valve, to the outflow 
of the products of combustion. 




That part of the diagram, which represents the effects due to 
throttling, has been purposely exaggerated. 

The curve DE is usually, more or less, wavy; this may be 
due to various causes. If the vibrations appear to be regular and 
of decreasing amplitude, they are principally due to the inertia of 
moving parts of the indicator. On the other hand, if the pressure 
is apparently constant for a time, then suddenly decreases, etc., 
the waves are due to friction between the piston and cylinder of 
the indicator. This trouble is readily removed by proper cleaning 
and lubrication of the piston and cylinder. Furthermore, waves 
may be established in the mixture in a manner similar to that 
described in Art. 69; i.e., as the piston begins to compress the 



IDEAL COEFFICIENT OF CONVERSION AND TESTS 217 

mixture a wave of compression travels through the medium to 
the other end of the cylinder, where it is reflected, with change of 
sign. This reflected wave then travels toward the piston; and 
when it meets the piston reflection again takes place, etc. In this 
manner, inequalities in pressure may be established, which under 
certain conditions may persist throughout the compression and 
expansion strokes. However, in general, these inequalities will 
not be manifested to any marked degree on the indicator diagram, 
since the inertia, of the moving parts of the indicator, will tend 
to suppress them. 

By an inspection of Fig. 41, it is obvious that the work done 
on the piston during the aspirating stroke is measured by the area 
JABK; and, likewise, the work done by the piston during the com- 
pression stroke is measured by the area KBCJ. During combus- 
tion, since there is no displacement of the piston, the work done 
is zero. During expansion the work done on the piston is measured 
by the area JDEFK. And, during expulsion, the work done by 
the piston is measured by the area KFGAJ. By taking the 
algebraic sum, we find that the net work done by the working sub- 
stance, during the cycle, is measured by the difference between 
the areas CDEFGi and AiB. 

Hence, if the mean effective pressure is determined by means of 
a planimeter, the tracing point of the planimeter, in tracing the 
area AiB must travel in a sense opposite to that pursued in tracing 
the area CDEFGi. That is, if i be the starting point, then, to 
find the difference between the two areas, the tracing point of 
the planimeter must follow, in order, the path i, C, D, E, F, G } 
i, A, B, i. 

The area AiB represents the work lost, due to valve throttling, 
and is, in well designed engines, small in comparison with the area 
CDEFGi. If the power lost, due to valve throttling, is large in 
comparison with the total indicated power, the valves must be 
readjusted. In general, the spring which gives good results for 
measuring the indicated power, has a modulus so high that the 



218 



THERMODYNAMICS 



part of the diagram, representing the power lost, due to valve 
throttling, is too small to be accurately measured. But, by using 
a stop, so as not to injure the spring, a much lower scale spring 
may be employed. In this manner the power lost, due to throt- 
tling, and also the amount of precompression may be accurately 
determined. 

167. Efficiency and Precompression. In Art. 161, equation 
(5), it was shown from theoretical considerations that, other 
things being equal, the thermal efficiency increases with the amount 
of precompression. This is found to be so in practice. There 
are, however, limits, beyond which the precompression may not 
be carried, due to the severe strains to which the engine is sub- 
jected during the explosion of the mixture. 

Tests made, in the Cooper Union Laboratories, on a Fair- 
banks 8 H.P. gas engine, gave the following results: 





R.P.M. 


B.H.P. 


Efficiencies. 


vjvz 


Thermal, 
Per Cent. 


Mechanical, 
Per Cent. 


4.72 
4.96 
5.09 
5.31 


368 
395 
414 

478 


6.54 
7.60 
7.92 
9.14 


13.9 
15.8 
16.1 
19.8 


74.0 
72.0 
68.0 
67.0 



The value given for the B.H.P. is, in each case, the maximum 
load the engine would carry for the given precompression. On 
attempting to carry the precompression higher than that given 
by i>i/fl2 = 5.31, it was found that the vibrations set up in the engine 
were so violent that satisfactory operation could not be obtained. 
From the table it is seen that the thermal efficiency increases 
rapidly with increased precompression. The mechanical efficiency, 
however, is considerably reduced. The fuel used during these 
tests was illuminating gas having a calorific value of about 590 
B.T.U. per cubic foot. The amount of precompression which, 
in any case, gives the best results depends, of course, upon the 
quality of the fuel used. 



IDEAL COEFFICIENT OF CONVEBSION AND TESTS 219 

It must, however, be emphasized that in any case, without 
considering the severe strains to which the engine is subjected, 
the amount of allowable precompression depends upon the tem- 
perature of ignition for the fuel used. For, if the temperature of 
the mixture due to the heat developed during the compression, 
becomes higher than that of ignition, premature explosions will 
occur, and the engine will not operate successfully. 

168. T-9 Diagrams and Internal Combustion Engines. The 
T-<p diagram, very frequently is a material aid in studying the 




Fig. 42. 



effect produced by a change in the cycle upon which an internal 
combustion engine operates. As an example, if we plot the T- 9 
diagram for a four-phase cycle, the effect produced by changing 
the amount of precompression is obvious from an inspection of 
the figure. Let A, in the T-cp diagram, Fig. 42, represent the 
condition of the mixture, as regards temperature and entropy, 
at the end of the aspirating stroke. Then, since the compression 
is assumed adiabatic, the entropy remains constant while the 
temperature rises from T4, that before compression, to T\, that 



220 THERMODYNAMICS 

after compression; the line representing this being parallel to 
the T axis, and the condition of the mixture, as regards temperature 
and entropy, is given by the point B. During combustion there 
is a rise in temperature, and also, an increment in entropy. The 
increment in entropy is given by 



9i 



X T 2 dT 
C.^r; (6) 



where M is the mass of the gas present, C v the thermal capacity 
per unit mass at constant volume, and T2 the temperature, when 
complete combustion has taken place. Though the thermal 
capacities of gases vary somewhat, for the ranges of temperature 
obtaining in an internal combustion engine, the variations are 
probably not very large. Hence, so far as the present discussion 
is concerned, no serious error is introduced by assuming C v 
constant, and equation (6) becomes 

91 = MC t log^ (7) 

The curve BC, therefore, representing the relation of temper- 
ature and entropy, during combustion, is logarithmic. During 
the expansion, which is assumed adiabatic, the entropy is con- 
stant, while the temperature falls from T2 to T%. Hence the 
curve, CD, representing this change, is parallel to the T axis. 
Finally, heat is rejected, the temperature falls from T3 to T4, 
and the relation of change in temperature to change in entropy 
is again logarithmic, as represented by the curve DA. By 
assuming the thermal capacity of the products of combustion 
constant, while the temperature falls from T3 to T4, we find for 
the change in entropy, 

92 = MC.logp. ...... (8) 

1 4 



IDEAL COEFFICIENT OF CONVERSION AND TESTS 221 

From Art. 161 we have 

T± TV 

hence, since T 2 is greater than Ti, T2 — T3 must be greater than 
T\ — T±, and CD on the diagram, must be greater than BA. 

Since, by Art. 109, the area under the curve, BC, is propor- 
tional to the heat absorbed, and the area under the curve, AD, 
is proportional to the heat rejected, it follows that the ideal coeffi- 
cient of conversion is 

Area FBCE- Area FADE Area ABCD , , 

T} ~ AresiFBCE ~ Area FBCE' ' ' W 

From equation (9), and by an inspection of Fig. 42, it is obvious 
that the ideal coefficient of conversion increases with increased 
precompression. Thus, if the precompression had been such that 
the temperature at the end of the compression were T2, as repre- 
sented by the point G, such that combustion takes place at the 
constant temperature I2, the ideal coefficient of conversion would 
be 

Area AGCD 



r} = 



Area FGCE 



which is obviously greater than that specified by equation (9). 
This is the principle upon which the Diesel motor operates; i.e., 
an attempt is made to bring about the application of heat at 
constant temperature. Again, if after complete combustion 
has taken place, the expansion be continued until the temperature, 
as represented by the point K, has been reached, the ideal coef- 
ficient of conversion is still further increased, and is given by the 
relation 

= AvesiAGCK 
71 Area FGCE ' 

which brings us back to the Carnot cycle. 



222 THERMODYNAMICS 

But, as has been previously explained, in Art. 115, this requires 
a stroke of greater length than is consistent with economy. 

169. Actual p-v and T-<p Diagrams of Internal Combustion 
Engine. The quantity of heat which a gas absorbs, or liberates, 
during a given temperature change, depends upon whether the 
change takes place at constant pressure or at constant volume. 
The change, however, usually takes place in a manner such that 
neither the pressure nor the volume remains constant. When 
both pressure and volume vary, the change in entropy is readily 
found from equations (42), (43), and (44), of Art. 49. Equation 
(42) states that 

dQ = C v dT--(C P -C v )dp. 

Assuming the process reversible, then, dividing through by T, 
we have 



and 



from which 



-—_clq-L p -Y—{L v — L v )—, 



r% n r T 2 dT m „. p ; 

I dy = C p \ -7p-(C p -C v ) I 



2 dp 
V 



92-9i = C,log|- 2 -(C,-a)log^; . . . (10) 
i i Vi 

where 91, T\, and p 1 represent, respectively, the initial entropy, 
temperature, and pressure, and 92, T2, and P2 represent, respect- 
ively, the final entropy, temperature, and pressure. 
Equation (43), of Art. 49, states that 

dQ = C4T+%{C v -C v )dv; 

from which, by substituting for p/R its value, T/v, and dividing 
through by T, we find 

dQ n dT , n n j&v 



IDEAL COEFFICIENT OF CONVERSION AND TESTS 223 
and 



Jc?! T Jti T ' J n v 



from which 



W -?i = C,logP+(C,-C.)log^. . . . (11) 

i 1 Vi 



Again, equation (44), of Art. 49, states that 

dQ = ^C v dp+^C P dv; 

from which, by substituting for v/R and p/R, respectively, 
T/p and T/v, and dividing through by T, we find 



dQ_j r dp dv 



and 



from which 



^9i ^pi p A v 



n-n = CAog^+C P \og v ^ (12) 

Pi Vl 

Equations (10), (11), and (12), then give the change in entropy, 
respectively, in terms of the change in temperature and pressure, 
the change in temperature and volume, and the change in pressure 
and volume. Equation (12), however, is usually the most con- 
venient; since, by means of a scale, after laying off on the indicator 
diagram, the axis of zero pressure and zero volume, the pressures 
and volumes corresponding to various points of the diagram are 
readily found; and from these, by choosing a suitable point for 
zero entropy, the entropy corresponding to the various points 
is easily computed. Dividing, equation (12), by C v , we have 

i(f 2 -?i)=log^+«log^; .... (13) 



224 



THERMODYNAMICS 



and since the scale employed, in plotting the T-y diagram is 
arbitrarily chosen, we may drop the factor 1/& and employ the 
equation 

... (14) 



92- 91 = log— +nlog-. 



Fig. 43 is a reproduction of an indicator diagram taken from 
an 8 H.P. Fairbanks gas engine, operating on illuminating gas. 
The ratio of the volume before compression to that after compres- 
sion, was 5.31; and the scale of the spring used, in taking the 
card, was 200 lbs. per square inch. 

H 




o v I 

Fig. 43. 

01 is the axis of zero pressure, obtained by measuring down, 
to proper scale, from the atmospheric line, AA' , a distance repre- 
senting the instantaneous barometric reading. OH is the axis 
of zero volume, obtained by measuring, to proper scale, to the 
left of the point A, a distance representing the equivalent length 
of the clearance volume. AB represents the aspirating stroke, 
BB' measures the rise in pressure, while the admission-valve is 
closing, B'C represents the compression stroke, CD represents 
the combustion, at practically constant volume, DE represents 
the expansion of the products of combustion, EFG represents the 
change from the time the exhaust-valve begins to open until 



IDEAL COEFFICIENT OF CONVEKSION AND TESTS 225 

it is fully open, and GJ represents the remainder of the expul- 
sion stroke. At the end of the expulsion stroke the exhaust- 
valve closes, the admission-valve opens, the pressure falls from 
J to A, and the cycle is completed. 

Fig. 44 is a representation of the T- 9 diagram, plotted from 
the p-v diagram, as shown in Fig. 43. Assuming the temper- 
ature at the point B, Fig. 43, to be equal to that of the atmosphere, 
and employing the characteristic equation, the temperatures 
corresponding to the various points were computed. An arbitrary 
value of entropy for the substance, corresponding to the point 




Fig. 44. 



where the curves, representing compression and expulsion, inter- 
sect, was assumed. Then by substituting in equation (14), the 
values of pressure and volume, as found by scale, from the 
p-v diagram, for the various points, the entropy, corresponding 
to these points, was readily computed. The value of n employed 
in making these computations, was a mean value, computed from 
the constitution of the mixture, before combustion, for the com- 
pression curve, and from the constitution of the exhaust gases, 
for the expansion curve. 

The point A, Fig. 44, represents the condition of the working 
substance, as regards temperature and entropy, at the point where 



226 THERMODYNAMICS 

the compression curve and expulsion curve intersect; and, the 
curve A B represents the relation between temperature and entropy 
from this point to the end of compression. BC is the curve, repre- 
senting the relation of temperature and entropy, for combustion, 
at practically constant volume. CDE represents the relation of 
temperature and entropy for expansion, and EA the relation of 
temperature and entropy for the abstraction of heat at, practi- 
cally, constant volume. 

Though, as has been previously stated, the thermal capacities 
of gases are not known for high temperatures, and therefore the 
diagram does not rigidly represent exact conditions, it still appears 
that, during compression, in this particular case, since the entropy 
is increasing as the compression advances, the cylinder walls are 
giving up heat to the mixture. From C to D the entropy of the 
substance appears, from the diagram, to be increasing, which 
is probably due to after burning; i.e., complete combustion has 
not taken place when the power stroke begins. From D to E 
the entropy first decreases and then increases, which appears 
to indicate that for a time, while the temperature of the products 
of combustion is high, heat is being given up by the gases to the 
cylinder walls, and later, as the temperature falls, heat is abstracted 
by the gases from the cylinder walls. In this manner, even 
though the T- 9 diagram may not represent exact conditions, 
conclusions may still be drawn, in regard to exchange of heat 
between the cylinder walls and the working substance. 

By assuming the compression adiabatic and computing the 
temperature for the point B, a value was found which was con- 
siderably less than that obtained, from Fig. 43, by means of the 
characteristic equation. This also shows that, during compres- 
sion, heat has been abstracted, by the gases, from the cylinder 
walls. 

170. Multi-cylinder Engines. Due to the fact that in a 
four-phase cycle there is only one power stroke for every four 
strokes, engines operating on this cycle are usually built with, 



IDEAL COEFFICIENT OF CONVERSION AND TESTS 227 

at least, two cylinders; but more frequently, especially for auto- 
mobiles, with four, six, and sometimes, even with eight cylinders. 
The explosions are so timed that the turning moment on the 
shaft throughout a rotation of the fly-wheel is, as nearly as pos- 
sible, uniform. 

171. Double-acting Cylinders. For power plants, internal 
combustion engines are, at the present time, frequently con- 
structed so as to be double acting; i.e., explosions properly 
timed, are brought about in both ends of the cylinder. However, 
for a well distributed thrust there must be two such cylinders, 
which may operate either tandem or twin. 

172. General Outline of Test. As stated in Art. 167, actual 
tests show that the thermal efficiency of an internal combustion 
engine is a function of the amount of precompression. Hence, 
in making a complete test of an engine, it is necessary to determine 
what precompression yields the best results. To do this, various 
methods may be employed; one, conveniently carried out, is 
that of fastening, by means of machine screws, disks of various 
thicknesses, and diameters equal to that of the cylinder bore, to 
the end of the piston, thus changing the clearance. In general, 
it will be found that, for every change in the amount of precom- 
pression, an adjustment of the governor is necessitated, so that 
the load is properly carried. At the same time, the calorific value 
of the fuel is determined, as well as the chemical constitution of 
the exhaust gases. If, by a chemical analysis, it is found that the 
exhaust gases show incomplete combustion, the amount of air 
admitted, during the aspirating stroke, must be changed, by a 
change in the admission valves, until an analysis of the exhaust 
gases shows complete combustion. Proceeding in this manner, 
step by step, it will be found that for every engine, there is a 
definite precompression which yields a maximal thermal efficiency; 
and, when this has been determined, the various losses are 
readily found. 

To determine the various losses, the load on the engine is 



228 THERMODYNAMICS 

maintained constant, for a given run, and noted. This gives the 
output of the engine. From the quantity of fuel consumed, 
samples of which are tested for calorific values, from time to time, 
during the run, the total energy consumed is found. The differ- 
ence between energy consumed, during the run, and the work 
delivered by the engine, during the same interval of time, con- 
stitutes the combined losses. The losses are chiefly: Mechanical 
losses, heat carried away by exhaust gases, heat carried away 
by the cooling water, heat lost by incomplete combustion, and 
heat lost, from the surface of the engine, by radiation and con- 
vection. 

The mechanical losses are determined by taking the difference 
between the indicated work and the work delivered. It is, of 
course, necessary to take a number of indicator diagrams, during 
the run, so as to obtain an average value for the indicated work; 
and furthermore, the number of power strokes should be deter- 
mined from the number of explosions, rather than from the num- 
ber of rotations, made by the fly-wheel. 

To determine the heat, carried away by the exhaust gases, 
the temperature of the gases, at the exhaust port, is determined 
by means of a pyrometer. And, from the constitution of the 
exhaust gases, as found by analysis, the mean thermal capacity 
per unit mass is found. Again, from the constitution of the fuel, 
the mass of fuel consumed, and the constitution of the exhaust 
gases, the total mass of the exhaust gases is readily computed. 

Finally, by taking the product of the difference in temperature 
between the exhaust gases and the room, the mass of the exhaust 
gases, and the mean thermal capacity per unit mass, the heat 
carried away by the exhaust gases is found. It must, however, 
be remembered that the thermal capacities for gases, as given in 
tables, are, in general, the results obtained by experiments con- 
ducted between the limits of 0°C. and 200°C; and, it is not at 
all certain that these values are correct for high temperatures. 

The heat carried away by the jacket water is determined 



IDEAL COEFFICIENT OF CONVERSION AND TESTS 229 

directly from the difference in temperature, between inflow and 
outflow, and the mass of water flowing through the jacket during 
the run. The temperatures are found by means of ordinary 
thermometers; and, the mass of water, by collecting in a suitable 
vessel and weighing. 

The heat lost, due to incomplete combustion, which should 
be very small, is computed from the constitution of the products 
of combustion. 

There finally remains, then, the heat lost by radiation and 
convection. This cannot be found directly, but is assumed to 
be equal to the difference between the input and the sum of the 
other losses plus energy delivered. 

Having found the various losses, a comparison may be made 
between the engine under test and other engines; and conclu- 
sions drawn therefrom in regard to making changes in the design 
or the method of operation. 



CHAPTER XVI 
COMPRESSED AIR AND COMPRESSORS 

173. Compressed air is used extensively and for a variety of 
purposes. It is used in tunneling; in mining, where, after it 
has done work upon an air motor, it may be employed for venti- 
lating purposes; in general, for power transmission; for air 
brakes on trains; etc. Hence, it is important that the compres- 
sion be brought about in the most economical manner possible. 
To do this, in attempting the design of an efficient air compressor, 
it is necessary to consider the underlying principles of thermo- 
dynamics, as well as those of machine design. 

However, before developing the formula representing the least 
amount of work that must be done in compressing a given mass of 
air, from one pressure to another, we will, as a matter of conve- 
nience, first determine the constants for air, and discuss briefly 
isothermal and adiabatic compression. 

174. Air Constants. The density of air, under a pressure of 
one standard atmosphere (1.01325 Xl.O 6 dynes per square centi- 
meter) and at a temperature of 32°F., is 0.001293 grams per 
cubic centimeter. Converting this to pounds and cubic feet, we 
find for the density, 0.08072 pounds per cubic foot; and, from this, 
a volume of 12.39 cu.ft. per pound. The pressure of one stand- 
ard atmosphere, expressed in gravitational units, is, approx- 
imately, 2115 lbs. per square foot; and, this corresponds, very 
nearly, with a pressure of 14.7 lbs. per square inch. Taking the 
product of pressure and volume, we find 

pv= 2115X12.39 = 26,200 lbs. per sq.ft. X cu.ft. 

230 



COMPRESSED AIR AND COMPRESSORS 231 

Since 32°F. corresponds, approximately, to 491.6 on the ther- 
modynamic scale, we find, from the characteristic equation, for 
1 pound of air, 

26,200 
H ~ 491.6 " 53,29 ' 

The foregoing constants have all been given to the nearest figure 
in the fourth place; but, in general, results of sufficient accuracy 
will be found by rounding off to the nearest figure in the third 
place. For, in designing compressors, some assumption has to be 
made regarding the average annual temperature of the atmos- 
phere ; and, frequently, it is assumed that this average temperature 
is 62°F. This, however, is not the proper value for all cases and 
localities. Assuming 62 °F. to be the proper temperature to be 
employed, then we shall have, for the product of pressure and 
volume, 

p aVa =RT a = 53.3X522 = 27,800; 

where T a is the temperature on the thermodynamic scale, corre- 
sponding to 62 °F., p a is the atmospheric pressure, and v a the 
corresponding volume, for 1 pound of air at 62 °F. 

175. Compression and Expansion. Assume the purpose, 
for which the compressed air is employed, to be that of driving 
an air motor; the construction of an air motor, and the cycle 
upon which it operates, being very nearly the same as that of a 
steam engine. Let it be further assumed that the compression, 
in the air compressor, takes place in a manner such that the heat 
developed, by the compression, is immediately absorbed by the 
surroundings; i.e., the compression is isothermal. The intrinsic 
energy of the air, then, at the end of the process, is precisely the 
same as at the beginning. Again, if it be assumed that the 
expansion in the air motor takes place in a manner such that the 
heat required during the expansion, in doing external work, is 
immediately supplied from the surroundings, as required, then the 



232 THERMODYNAMICS 

intrinsic energy of the air, at the end of the process, is precisely 
the same as at the beginning. And under these assumed condi- 
tions, the work done on the air, while being compressed, is pre- 
cisely equal to the work done by it, while expanding, between the 
same limits of pressure. It is thus seen that a vessel, containing 
compressed air, at room temperature, does not constitute a reser- 
voir of energy; but, the air is merely in a condition such that it 
can absorb energy, in the form of heat, from the surroundings, 
and convert it into mechanical work. On the other hand, if the 
compression is adiabatic and the vessel, in which the compressed 
air is stored, is perfectly insulated, such that the heat developed 
during compression is retained, then the vessel does constitute 
a reservoir of energy; for, by an adiabatic expansion, the work 
done on the air, during compression, may again be recovered; and 
the air at the end of the expansion is in precisely the same con- 
dition as it was at the beginning, without any exchange having 
taken place between the air and the surroundings. This, of course, 
assumes no other losses. The receiver, then, in the latter case, 
containing the compressed air, constitutes a reservoir of energy, 
not because it contains compressed air, but merely because heat 
has been stored which may be reconverted into mechanical work. 
In general, however, when air is compressed, the compression 
is very nearly adiabatic; and the air is stored in a receiver, 
where, in a very short interval of time, the heat developed 
during compression is, by means of conduction and radiation, 
given up to the surroundings, and is irrevocably lost. Hence, 
when expansion takes place in the motor, which is again, approx- 
imately, adiabatic, the external work is done at the expense 
of the intrinsic energy of the air; and, accordingly, the tem- 
perature falls. Hence, since the intrinsic energy of the air is 
practically a function of the temperature only, it follows that 
the intrinsic energy at the end of the expansion, in the motor, 
is less than it was initially in the compressor, at the instant 
when compression began. 



COMPRESSED AIR AND COMPRESSORS 



233 



176. Isothermal Compression and Expansion. The results 
just deduced, in Art. 175, are best illustrated by means of the 
ideal p-v diagram; first, by considering isothermal processes, and 
then by considering adiabatic processes. Let, in Fig. 45, 01 
and OH be, respectively, the axes of zero pressure and zero 
volume; and, for the present discussion, it will be assumed that 
the compressor has no clearance. The work done, then, on 
the piston by the air, during the aspirating stroke, as represented 
by the line AB, is measured by the area OABE. During the 
isothermal compression, represented by the curve BC } the work 




done by the piston, on the air, is measured by the area EBCF. 
At the point C, the air is under a pressure equal to that obtaining 
in the receiver, the exhaust-valve opens, and the air under a 
constant pressure, as represented by the line CD, is forced into 
the receiver, and the cycle is completed. The work done by 
the piston on the air during expulsion, is measured by the area 
FCDO. Finally, the net work done by the compressor, during 
the cycle, is obviously measured by the difference between the 
areas EBCDO and EBAO; i.e., the area ABCD. 

Representing the pressure and volume of the air, corresponding 
to the point B, respectively, by p a and v a , and likewise, for the 
point C, by pi and vi, then, by considering the work done on 



234 THERMODYNAMICS 

the piston negative, and the work done by the piston positive, 
we have for the work done, during the aspirating stroke, 

Wi = -p a va. . (1) 

The work done, during the compression, BC, is 

W 2 =- C'pdv; (2) 

J Pa 

and the work done, during expulsion, is 

W 3 = PiV! (3) 

Finally, the net work done, during the cycle, is the algebraic 
of Wi, W2, and W3; i.e., 

W=—p a Va—\ pdv+pivi; (4) 

J Pa 

where W is the net work done. But, since the compression is 
isothermal, 

PaV a = PlVi; 

and equation (4) becomes 



W=- pdv (5) 

J Pa 

Substituting now, in equation (5), for dv its value, as found from 
the equation 

pV = p a V a , 

we find 

W = Pa Vaf Pi ^ = PaVal0 g ^ (6) 

JVa P Pa 

If we are dealing with 1 pound of air, and assume the tem- 
perature, during the aspirating stroke, to be 62 °F., then we may 



COMPRESSED AIE AND COMPRESSORS 235 

substitute, in equation (6), the value of p a v a as found in Art 
174, and we find 

W = 27,800 log 2± f t.-lbs. per pound. ... (7) 



That is, the right-hand member of equation (7) is the expres- 
sion for the amount of work in ft. -lbs., that must be done in taking 
in, under the assumed conditions, 1 pound of air at a pressure 
p a , compressing it to a pressure pi, and forcing it against this 
pressure into a receiver. It is obvious that the ratio pi/p a is 
independent of the unit of measure chosen; but, it is necessary, in 
order to obtain the work in ft. -lbs., that the numerical coefficient, 
in equation (7), be deduced by expressing the pressure in lbs. per 
square foot and the volume in cubic feet. 

Assume, now, that the process takes place in the reverse 
order, step by step, in a manner such that the diagram is traced 
in the order, DCBA. By assuming a perfect regenerator, such 
as was discussed in Art. 27, the process becomes ideally reversible, 
and the net work done, on the motor, is precisely equal in 
amount to that done by the compressor, as given by equation 
(6). And since this is the best that can be done in any case, 
the indicator diagram, representing isothermal compression, is 
the one chosen as a standard, for the comparison of the per- 
formance of air compressors. 

177. Adiabatic Compression. For the same initial and final 
pressures, the work done, during compression, and also during 
expulsion, will not be the same for an adiabatic process as it is 
for an isothermal process. On the other hand, the work done 
during the aspirating stroke is the same in either case. 

Let in Fig. 46, the line A B represent the aspirating stroke, 
the curve BE the adiabatic compression, and the line ED the 
expulsion stroke at the pressure of the receiver. The net work 
done by the piston is measured by the area ABED. Had the 
compression been along the isotherm BC, the net work done 



236 



THERMODYNAMICS 



would be measured by the area ABCD; hence, the excess of work 
done, during a cycle, when the compression is adiabatic, over 
that done when the compression is isothermal, is measured by 
the area BCE. 

The work done during the aspirating stroke is again 



TPi 



VaVa) 




(8) 



where the symbols have the same significance as in Art. 176. 
The work done, during adiabatic compression, is 



Wi 



rvx 
- I pdv: 



(9) 



but, since we are now dealing with an adiabatic change, the 
value of dv to be substituted in equation (9) must be deduced 
from the equation 



Solving for v we find 



pv n = p a Va n . 



1 _1 

V = p a n V a p n , 



from which, 



1 I l+w 

dv= Pa n V a p » dp. 

n 



COMPRESSED AIR AND COMPRESSORS 237 

Substituting in equation (9), we find 



n JVn 



1_ 

p a nV a/ 2=1 2=1 

= ^^(P1 » "Pa » ) 

-stfer-i <>»> 

Since now the volume, at the end of the adiabatic compression, is 

i_ _i_ 

Vi = Pa n V a pi », 

the work done, during expulsion, is 

n-l 
1 2zi /©A n 

W3 = PiV 1 =p a n V a pi n =PaVal—J . • (11) 

Taking the algebraic sum of the right-hand members of equa- 
tions (8), (10), and (11), we find, for the net work done by the 
compressor, 

n— 1 n— 1 

which reduces to 

n-Y [\p a J j 

It was stated, in Art. 176, that the cycle having isothermal 
compression may be taken as a standard cycle. The ratio of 
the work done, during a cycle, when the compression is iso- 
thermal, to that when the compression is adiabatic, may be 
termed the theoretical efficiency of compression. Dividing equation 



238 



THERMODYNAMICS 



(6) by equation (12), we find, for the theoretical efficiency of 
compression, 



log 



Y] = 



n-l 

n J Ip 
n-l] \p a/ 



(13) 



As a matter of convenience a table is here given, which was 
obtained by computing the theoretical efficiencies, by means of 
equation (13), on the assumption that n has the value of 1.4. 



Pi 

Pa 


Per Cent. 


El 

Pa 


Per Cent. 


1.5 


94.3 


6 


76.6 


2 


90.4 


7 


74.8 


3 


85.1 


8 


73.2 


4 


81.5 


9 


71.9 


5 


78.8 


10 


70.7 



It will be noted that, when the ratio pi/p a = 3, the theoretical 
efficiency of compression is approximately 85 per cent, and when 
pi/p a = / l, it is approximately 81.5 per cent. Hence, when the 
ratio of pi to p a is greater than 3 or 4, the losses, from a ther- 
modynamic standpoint, become excessive. Therefore, if it be 
desired to operate economically, it becomes necessary to limit 
the ratio of final to initial pressure to a value between 3 and 4. 

Equation (12) may be transformed so as to express the work 
done by the compressor, in terms of the initial and final tem- 
peratures. Since 



n-l 
Ta \Pa) 



where T a is the initial and T\ the final temperature, and since 
p a Va = BTa, 



COMPRESSED AIR AND COMPRESSORS 239 

we find, by substituting in equation (12), 

w =^i RT ^¥ a - 1 ) (14) 

And finally, by substituting, in equation (14), for n and R, 
respectively, the equivalent values, C P /C V , and J(Cp—C v ), we 
find 

W = JC P (T 1 -Ta) (15) 

That is, equation (15) shows that the net work consumed by an 
air compressor, per cycle, per pound of air, when the com- 
pression is adiabatic, is precisely equal in amount to the heat, 
expressed in mechanical units, consumed in elevating the temper- 
ature of the air, at constant pressure, from that before compres- 
sion to that after the compression is completed. 

Equation (15) may be deduced in an entirely different, though 
very simple, manner. The work done by the piston, during 
admission, is 

W 1 =-p a v a =-RT a =-J(C p -C v )T a . ... (a) 

And, during adiabatic compression, since the intrinsic energy of 
the air is a function of its temperature only, the work done by 
the piston is 

W 2 = C V (T 1 -T a ) .... (6) 

The work done by the piston, during exhaust, is 

Ws = Vl v 1 =RT 1 =J(C p -C v )T 1 ; ( c ) 

taking the sum of Wi, W2, and W3, as given by equations (a), 
(6), and (c), we find 

W = JC P (T 1 -T a ). 

178. Multi-stage Compression. Various methods, in which 
an attempt is made, to bring about compression, approaching 



240 THEKMODYNAMICS 

an isothermal process, have been tried; but, it appears impossible 
to reduce the exponent, n, to a value approaching unity. The 
various methods used are: Water-jacketing, playing a jet of 
water into the cylinder while compression is taking place, and 
spraying, by means of an atomized jet of cold water, the air while 
it is being compressed. Water-jacketing appears, so far as the 
results of investigations show, to give very little, if any gain in 
economy. That is, when recourse is had to jacket cooling, the 
heat developed by compression is absorbed so slowly that the com- 
pression is practically adiabatic. When the cooling is attempted 
by means of a jet of water, played into the cylinder, the exponent, 
n, may be reduced to a value of about 1.35; and when the 
cooling is brought about by an atomized spray, the value of n 
may be reduced to about 1.25. Hence, at best, the compression 
is far from approaching an isothermal process; and, for efficient 
operation, when the ratio of the final pressure to the initial 
pressure is greater than 4, recourse must be had to multi-stage 
compression. 

We will consider, first, a two-stage compressor. That is, 
during the aspirating stroke, a certain quantity of air, at a 
pressure p a , flows into the cylinder, which on the return stroke 
is compressed to some pressure p±; the relation of pressure 
and volume being given by the equation 

pv n = k ; 

where the exponent n, depending upon the method of cooling 
applied, may have a value ranging from about 1.25 to 1.4. At 
the end of this compression stroke, when the pressure pi has 
been attained, the air is expelled into a receiver under a con- 
stant pressure p\. The receiver has a jacket through which 
water is circulated in a manner such that the heat developed, 
during compression, is removed; and the product of pressure 
and volume, after cooling in the receiver, is equal to the product 
of pressure and volume at the instant the compression began. 



COMPRESSED AIR AND COMPRESSORS 



241 



In other words, the condition of the air, as regards pressure and 
volume, in the receiver, is the same as though the compression 
had been isothermal; i.e., 

PlV%= PaVa. 

Let, in Fig. 47, the line AB represent the aspirating stroke, 
for the low-pressure cylinder, and the pressure and volume, 
corresponding to the point B, be given, respectively, by p a and 
v a . The compression then takes place approximately adiabat- 



err-- f 




Fig. 47. 

ically, as represented by the curve BE. At the point E the 
exhaust-valve opens and expulsion takes place, to the receiver, 
at a constant pressure pu The volume of the receiver being large 
in comparison with that of the first cylinder, the pressure in it 
is sensibly constant; and the volume of air, in cooling from T\, 
the temperature at the end of the compression BE, to T a , the 
atmospheric temperature, shrinks from the volume, represented 
by FE, to that represented by FG, such that 

Pivi=p a v a ; 

and the point G is on the isotherm through the point B. For the 
same quantity of air, then, the line FG represents the aspirating 



242 THERMODYNAMICS 

stroke for the second, or high-pressure, cylinder. The curve GJ, 
represents the compression, which is practically adiabatic, to the 
pressure p2, existing in the reservoir in which the air is stored. 
Finally, the expulsion stroke to the reservoir is represented by the 
line JD. 

The various quantities of work involved during the cycle are 
as follows: During the aspirating stroke, for the low-pressure 
cylinder, the work done on the piston is measured by the area 
ABKO; and during compression, to the pressure pi, the work 
done by the piston is measured by the area KBEL; and during 
expulsion, to the receiver, the work done by the piston is measured 
by the area LEFO. Hence, the net work done by the piston, in 
the low-pressure cylinder is measured by the area ABEF. In a 
similar manner, we find that the net work done, by the piston in 
the high-pressure cylinder, is measured by the area FGJD. The 
total net work done, therefore, by the two-stage compressor, 
during the cycle, is measured by the area ABEGJD. Had the 
compression taken place in a single-stage compressor, between the 
same limits of pressure, the net work done, by the piston, would 
be measured by the area ABEND; where the curve BEN repre- 
sents an adiabatic through the point B. Hence, the saving in 
work, neglecting losses, by employing a two-stage compressor, is 
measured by the area ENJG. And the work done by the piston 
of this two-stage compressor, in excess of that which would have 
been done had the compression been isothermal, is measured by 
the sum of the areas GBE and CGJ, where the curve BGC repre- 
sents an isotherm. 

By equation (12), we have for the work done in the low-pres- 
sure cylinder, per cycle, when 1 pound of air is taken in at a pres- 
sure p a , is compressed adiabatically to a pressure p\, and expelled 
at this pressure to a receiver, 

"'^•-{(gf - 1 ! <i6) 



COMPEESSED AIR AND COMPRESSORS 243 

In a similar manner, the work done in the high-pressure cylinder, 
when 1 pound of air is taken in at a pressure pi, is compressed 
adiabatically to a pressure p2, and expelled at this pressure to 
a receiver, is 

Since, however, the air in the intermediate receiver has its tem- 
perature reduced to the initial value, we must have 

and equation (17) becomes 

^nh^W- 1 ] (18) 

Taking the sum of equations (16) and (18), we find that the total 
work done, during a cycle, by the two-stage compressor, is 



n — l 



,$ +© - 2i - « 



Now, p a is a constant and so is p2', since, for any particular 
case, p2 is the desired final pressure, pi, however, is a variable; 
and the value of W obviously depends upon the value chosen for 
pi. Since, the only variable in the right-hand member of equation 
(19) is pi, the value found for W is a minimum when the expression, 

n— 1 n—l 

p.) + \n) ~ 2 ' 

is a minimum. Differentiating this expression, with respect to 
pi, equating to zero, and solving for p\, we find 

Pi = pA>2* (20) 



244 



THEEMODYNAMICS 



Substituting the value of pi as given by equation (20), in equations 
(16) and (18), we find 



and, 



*-£H (£»*-' 



» .-V>.((e)"*"'-i 

n-r [\Pa 



. . . (21) 



(22) 



From equations (21) and (22) it is seen that, if the work done, 
during a cycle, by a two-stage compressor, is to be a minimum, 
it must be equally divided between the two cylinders. Taking 
the sum of the right-hand members of equations (21) and (22), 
we find the net work done, when employing the most efficient 
compression possible, by a two-stage compressor, in taking air 
under a pressure p a and expelling to a receiver, under a pressure 
V2, is 

n-l 

W = ~iPaV a { (-) ^ - 1 1 ft.-lbs. per pound. . (23) 



If the compression is brought about by three stages, the final 
pressure being p%, and the pressures of the intermediate receivers, 
respectively, p\ and p2, then, on the assumption that, in the two 
intermediate receivers, the temperature is reduced to that of the 
atmosphere, the work done in the first, second, and third cylinders 
is given, respectively, by 



and 



Wi = 



n—1 



PaVa 



W2 = -- 1 PaV, 



n — 1 



n-l 

n 

Pa 

n- l 

^ n 

Pi. 



n-l 



(24) 
(25) 

(26) 



COMPRESSED AIR AND COMPRESSORS 245 

Taking the sum of Wi, W2, and W3 we find, for a cycle, the net 
work done by the three-stage compressor, is 

The right-hand member of equation (27) is a minimum when the 
expression included in the brace is a minimum. Differentiating 
this expression, first, assuming pi variable, and p a , p2, and pz 
constant, equating to zero, and solving for pi, we find 



Pi 



= Vp^P2 (28) 



Equation (28) gives the relation of p\ to p a and p2 such that the 
process in going from p a to P2 shall involve a minimum amount 
of work. Differentiating again, this time, however, assuming 
pi and p3 constant, and P2 variable, in order to obtain the relation 
2?2 must bear to pi and p% such that a minimum amount of work 
is involved, while the process takes place from pi to p%, we find 

V2 = ^pm (29) 

By elimination we find, from equations (28) and (29), 

vi=^vln, (30) 

and 

V2 = </^1- (31) 

Substituting, in equations (24), (25), and (26), the values of pi 
and p2, as given by equations (30) and (31), we find 

w-l 

"^ v-lfe)"- 1 ! < 32 » 

"■-sM©*-'} <33) 

and 

n-l 



"-iM©""- 1 ) <34 » 



246 THERMODYNAMICS 

Equations (32), (33), and (34), again show that, for the most 
economical compression, the work must be equally divided between 
the cylinders; and the net work, for the three-stage compressor, is 

W=^~VaV a I (-) ^ - 1 1 ft.-lbs. per pound. . (35) 

In a similar manner, for the most economical four-stage com- 
pressor, we find 



ra-l 

TJ7 4n f fpA 4« 

W = --.VaVa\ — ) "I 

n-V [VpJ 



ft.-lbs. per pound. . (36) 



By a comparison of equations (12), (23), (35), and (36), it is 
seemthat in each case the coefficient n/(n — 1), is multiplied by the 
number of stages, and the exponent {n—l)/n, is divided by the 
number of stages; and in all other respects, the equations are 
identical. We may then write a general equation for the net 
work done by a multi-stage compressor, in taking in 1 pound of air 
under a pressure p a , compressing it, by means of S stages, to a 
pressure p, and expelling it, at this pressure to a reservoir. This 
net work done is given by 

n-l 

W = ^ n -p a va I (~ ) ^ ~ 1 1 ft.-lbs. per pound. . (37) 
n-V [\p a / j 

It is obvious that for any two given initial and final pressures, 
the greater the number of stages, the smaller the areas, representing 
the difference in work, between adiabatic and isothermal compres- 
sion, become; and hence, the nearer the compression approaches 
an isothermal process. And, in the limit, as S in equation (37) 
becomes indefinitely large, the compression becomes isothermal. 
However, increasing the number of cylinders, increases the bulk 
and first cost of the compressor, as well as the loss of work due 
to friction and imperfect valve action. On the other hand there 
are also certain mechanical advantages in a multi-stage com- 
pressor, similar to those discussed, in Arts. 141 and 144, for the 



COMPRESSED AIR AND COMPRESSORS 



247 



compound engine. That is, for a two-stage compressor the 
stresses in the moving parts are less than in a single-stage com- 
pressor operating between the same limits of pressure; and, if 
the compressor be cross-compound, the thrust on the crank bear- 
ings is more uniform. But, in any given case, there is a limit- 
ing value for the number of stages; practically when the thermo- 
dynamic gain is offset by the interest on the extra capital invested, 
plus depreciation and mechanical losses. For final pressures 
of about six atmospheres, two-stage compressors are usually 
employed. 

179. Clearance. The clearance of an air compressor is very 
small in comparison with the clearance of a steam cylinder. Still, 




account must be taken of the clearance in designing a compressor; 
since, due to it, the effective volumetric displacement of the piston 
is less than the actual volumetric displacement. Let, in Fig. 
48, AB represent the displacement of the piston, BC the com- 
pression, CD the expulsion, and EA the clearance. Then, since 
the volume of ah,EA, remaining in the cylinder, when the exhaust- 
valve closes, is under a pressure pi, expansion must take place to 
atmospheric pressure, as represented by the curve DF, before the 
inlet-valve opens and a new supply of air flows into the cylinder. 
Hence, the effective piston displacement is measured by FB. 



248 THERMODYNAMICS 

Let K * be the ratio of the actual piston displacement, AB, 

to the clearance, EA, then 

A R 
EA = ~ (38) 

For the expansion along the curve DF, we may write 

VlVi n = VaVa n ; (39) 

where v\ and v a are the volumes, respectively, as represented by 
EA and EF. From equation (39), we find 

i 



Vi\ n 



Pc 

and, by substituting for v\ and v a , respectively, the value of EA, 
as given by equation (38) , and EF, we obtain 

— ¥©* (»> 

Now, the effective displacement is given by 

FB=AB-(EF-EA); (41) 

hence by substituting, in equation (41), the value of EF, as given 
by equation (40), and the value of EA, as given by equation (38), 
we find 

The effective piston displacement not being equal to the actual 
piston displacement, does not affect the expression deduced for 
the work done on an air compressor; for, the air remaining in the 
cylinder, at the end of the expulsion stroke, does an amount of work 

* K usually has a value of about 50. 



COMPRESSED AIR AND COMPRESSORS 249 

on the piston, in expanding, which is practically equal to that 
which was done on it while being compressed. The effect of the 
clearance, then, is merely to reduce the capacity of the cylinder. 

180. Throttling and Other Imperfections. The capacity of a 
cylinder of an air compressor is very frequently more seriously 
affected by other causes than it is by clearance. In the first place 
there is always, due to valve friction, a certain amount of throt- 
tling, which causes the pressure in the cylinder, during the aspira- 
ting stroke, to be less than atmospheric. Further, due to imper- 
fect valve action, i.e., the valves not opening or closing at the proper 
time, the capacity is reduced. And, finally, the temperature of 
the cylinder walls is usually higher than that of the incoming air, 
which again tends to reduce the capacity of the cylinder. These 
combined causes may reduce the apparent capacity, depending 
upon the speed of the machine, from 5 to 20 per cent. 

181. Adiabatic Expansion in Motor. The cycle of an air 
motor is practically the reverse of that of an air compressor. The 
admission-valve opens and air from the mains, under a practically 
constant pressure, forces the piston forward to the point of cut- 
off, and the work done on the piston, per pound of air is 

Wi=jpivi) (43) 

where pi is the pressure in the main, and vi the volume of one 
pound of air at cut-off. The expansion is then practically adia- 
batic, and the work done in expanding from the pressure pi, to 
p a , that of the atmosphere, is 

n-l 

(44) 



w f Pa a l ~ C Va ~-j VVJi 
W2= I pdv= p\ n vi I p n dp = I - — - 

Jvx n Jpi n ~ l 



!_i* 



Pi 



The exhaust-valve then opens, the air is expelled under a pressure 
p a , and the work done, by the air, is 

1_ n-l 

W3=-PaVa=—pi n Vip a « . . . ' . . . (45) 



250 



THERMODYNAMICS 



Taking the sum of the right-hand members of equations (43) 
(44), and (45), we find, for the net work done on the motor 
per pound of air, 



n-l 



w=- p - lV -\< > 

n — 1 



r + * 



n-l 



n 1 (Pa 

n-r \pi 



1_ n-l 

Wi — pinViPa n 



ft.-lbs. 



(46) 



Since the temperature in the mains is practically atmospheric, 
pivi is the product of pressure and volume, for one pound of air 
under ordinary conditions, and may be replaced by the constant 
27,800. Hence, equation (46) becomes 



n-l 



IF = 27,800— — \ 
n—1 



(47) 



The indicator diagram for the preceding discussion is shown 
in Fig. 49, in which AB represents the admission, BC the expansion, 




and CD the expulsion. The work done on the piston during 
admission and expansion, is measured by the area ABCGF; and 
the work done by the piston, during exhaust, is measured by the 
area CGFD. Hence, the net work done, by the air, is measured 
by the area ABCD. 



COMPRESSED AIR AND COMPRESSORS 251 

Equation (46) may be put into another form, by substituting 

n-l 

for pivi, its value RT\, and for (p a /pi) n , its value T a /T\. 
Making these substitutions, we find 

from which, for adiabatic processes, since in that case n = C P /C V , 
we obtain 



And, since 

we have finally 






R = J(Cp — Cv), 

W = JC P (T 1 -T a ) (48) 



It must be noted that, in equation (48), T\ is the temperature in 
the mains, which is practically that of the atmosphere, and T a , 
the temperature of the air after expanding adiabatically from the 
pressure pi, that existing in the mains, to p a , that of the atmos- 
phere. 

182. Reheating. When air at atmospheric temperature, 
and under a high pressure pi, expands to atmospheric pressure p a , 
the corresponding temperature, T a , will be very low. As an exam- 
ple, if air under a pressure of five atmospheres, and at atmospheric 
temperature Ti, expands adiabatically to a pressure of one atmos- 
phere, its temperature becomes approximately, 

n-l 2_ 

T a = Ti(^\ n =522^|V =330=-130°F.; 

n having been assumed to have the value 1.4. Temperatures 
as low as this, due to the fact that the moisture present in the air 
freezes, makes lubrication difficult and clogs the valves, are 
undesirable at the exhaust of an air motor. To prevent this, 



252 THEKMODYNAMICS 

recourse must be had to reheating; i.e., the air from the mains 
is passed through a heater before being admitted to the motor. 

In being heated at constant pressure, the volume of the air 
is increased, and the ratio of the two volumes is given by T/Ti; 
where T is the temperature of the air after heating. Hence, 
equation (46) , giving the work done, per pound of air, on the air 
motor, becomes 



=^x'- n -^i{i-(|) n \ 



Tin-V 



If, now, T a r is the temperature at the end of the adiabatic expansion, 
to the pressure p a , and since n = C P /C V , pivi =RT\= J(C P — C v ) Ti, 

n-l 

and (pa/pi) n = Ta'/Ti, we find, by substituting in equation (49), 

Wl= ¥ 1 x c^/ iCp ~ Cv)Tl { 1 ~^)' ' ' (50) 

And further, since the ratio of final to initial pressure is the same 
whether there be reheating or not, the ratio of final to initial 
temperature must also be the same for both cases; hence, 

T 

rp t rp±J* . 

J-a — -L m ) 

where T a is the final temperature when there is no reheating. 
Substituting this value of T a ' in equation (50), and simplifying, 
we find 

Wi-JcSiTi-T.) (51) 

J- 1 

Equation (51) is the expression, in terms of the three temper- 
atures, with reheating, for the work done per pound of air, on the 
air motor. Equation (48) is the expression for the work done per 
pound of air without reheating. Taking the difference between 
equations (51) and (48), we find, due to reheating, for the gain 
in work 

W' = JC v ~(T l -T a )-JCATi-T a )=JC v (T 1 -T a ) T -~±. (52) 



COMPRESSED AIR AND COMPRESSORS 253 

The heat consumed, expressed in mechanical units, in raising 
the temperature of 1 pound of air at constant pressure, from T\ to 
T, is 

and the work which could be realized from this quantity of heat, 
by means of a Carnot cycle, is 

W" = JC V {T-T 1 ) T ^-. ..... (53) 

Taking the ratio of W, as given by equation (52), to W", as given 
by equation (53), we find 

W'_T Ti-Tg 
■\ty" Ti T—Ti ^ ' 

In equation (54), the first factor, viz, T/Ti, is always greater 
than unity, and for any given case, T\ — T a is a constant. Hence 
the ratio, W ' /W" ', is greater than unity until the air is reheated 
to a temperature such that 

^ = ^A (55) 

And, for reheating to a temperature higher than this, the ratio 

becomes less than unity. Solving equation (55), for T, we find 

T 2 
T = ~r (56) 

Hence, for reheating to temperatures lower than that given by 
equation (56), there is a thermodynamic gain; i.e., the gain in 
work, due to the heat applied in reheating the air, is greater than 
that which could be realized if an equal quantity of heat were 
utilized on a Carnot cycle for the same limits of temperature. 

To illustrate, we will assume a particular case and solve for 
T. Let pi, the pressure in the mains, be six atmospheres, T\ be 
522, and p a , the final pressure, be one atmosphere; then 

M ter= 522 (^ f 



254 THERMODYNAMICS 

From equation (56), we find 

_ T x * (522)2 



5- = 871 + . 



522( 6 - 

Giving a temperature for reheating, above that existing in the 
mains, of practically 349° F., which is higher than ever employed 
in practice. 

The heat consumed in reheating is applied to much better 
advantage than in the case of a steam engine and boiler. Further- 
more, since the fuel used in reheaters may be of a much lower 
grade than that ordinarily employed for heat motors, there is a 
saving in cost of fuel. 

183. Loss of Head in Transmission Pipes. When a liquid 
flows in a pipe, there is always, due to friction between the liquid 
and the surfaces with which it comes into contact, a resistance 
to be overcome; and on account of this, there is a loss in pressure. 
That is, when friction is taken into account, Bernouilli's Theorem, 
which states that, for the steady flow of a liquid in parallel stream 
lines without friction, the pressure head plus the velocity head plus 
the static head is a constant for any section under consideration, 
no longer applies. The energy consumed in overcoming fric- 
tion, is manifested by eddy currents. These eddy currents in 
turn subside; and the energy, possessed by them, is converted into 
heat, which in turn is lost by being dissipated to the surroundings. 
The loss of energy thus experienced by a given mass of the liquid 
is usually expressed by a loss of head. That is, the loss of head, 
experienced by a unit mass of the liquid, is numerically equal to 
the vertical height through which it would have to fall to do an 
amount of work equal to that consumed in overcoming the 
friction. 

From a great number of experiments upon the flow of liquids 
in pipes, the following facts have been adduced : The loss of head 
is very nearly proportional to the square of the speed of flow, 



COMPRESSED AIR AND COMPRESSORS 255 

varies directly as the length of the pipe, as the wetted perimeter, 
and inversely as the cross-sectional area of the stream. 
These relations may be stated symbolically as follows : 

H =f s if' < 57 > 

where / is an experimental constant depending upon the nature of 
the liquid and inner surface of pipe, and H, s, L, P, and A are, 
respectively, the loss in head, speed of flow, length of pipe, wetted 
perimeter, and area of stream. For any particular cross-section 
the ratio of A to P is a constant; which is termed the hydraulic 
radius, and may be replaced by the symbol K. Hence, equation 
(57) may be written 

H =fSk ^ 

Since the temperature of air, flowing in a pipe of any consider- 
able length, is sensibly constant, the product of pressure and volume 
is also practically constant; and hence, as the pressure falls the 
speed of flow must increase. Therefore, since equation (58) 
assumes a constant speed, it is not directly applicable to the flow 
of air, or any other gas. In the limit, however, we have 



dH=f~dL. ....... (59) 

And, since the loss of head is numerically equal to the work done 
by a unit mass of the substance, we have 

dH = pdv; (60) 

where p is the pressure, and dv the change in volume, per unit 
mass, for the section under consideration. From equations 
(59) and (60), we find 

pdv=f£ K dL (61) 



256 THERMODYNAMICS 

There is, of course, due to change in speed, also a change in 
kinetic energy; but, in general, this is so small in comparison with 
the total loss of head that it may be neglected. 

Substituting, in equation (61), for dv its value as obtained 
from the equation 

pv = RT, 
we find 

f dp= - f m dL • • (62) 

Under steady flow the mass of air passing any section, for a 
given interval of time, is a constant throughout the entire length 
of pipe. Hence, we have, for the speed of flow, 

s= ~a=^pa> (63) 

where M is the mass passing any section per unit time, v the volume 
per unit mass, and A the cross-sectional area of the pipe. Sub- 
stituting the value of s, as given by equation (63), in equation 
(62), we find 

, M 2 RT „ 

pdp= ~ f 2gKA^ dL; 
from which 

)j dv= - f w^) dL; (64) 

where pi and p2 are, respectively, the initial and final pressures, 
and L the length of the pipe. Finally, integrating, as indicated 
in equation (64), we find 

v*-*-*^ <«> 

From equation (63) we have 

Vl sM 2 '' (66) 



COMPKESSED AIR AND COMPRESSORS 257 

where si is the initial speed. Dividing equation (65) by equation 
(66), member by member, we find 

Pi 2 ~P2 2 =f sJL 

pi 2 ] gKRT { °° 

Solving equation (67), respectively, for p2, si, and/, we find 

»-»(?-{gm?: (68) 



Sl 

and 



K^^)' « 



'^>m <™ 

By means of equations (65), (66), (68), and (69), the necessary 
calculations, for any given case, may be made; and, by means of 
equation (70), the coefficient / may be found for a given set of 
observations. 

The ratio A/P is, for cylindrical pipes, a function of the diam- 
eter only; i.e., 

xP 2 /4 ^D 
%D 4* 

We may substitute, then, in equation (68), the following con- 
stants: = 32.2, K=D/4:, and # = 53.3, and find 

f Sl 2 L V /i M 2 L \i 

*-* 1 - Mx .wxfr) ■"(-«»)■ <7 " 

In a similar manner, the various equations may be simplified. 

Equation (71) is, perhaps, best illustrated by assuming a 
concrete case, and solving for the terminal pressure. As an 
example, let it be required to find the final pressure, for the case 
when the initial pressure is six atmospheres, the temperature 



258 THERMODYNAMICS 

62 °F., the quantity of air required 1200 cu.ft. per minute, the 
length of pipe 5 miles, and the diameter of the pipe is 1 ft. 
First of all, from equation (71), it is obvious that the pressure 
may be specified in any units whatsoever. 

From a series of observations made by Riedler and Gutter- 
muth upon the compressed air system of Paris, extending over a 
distance of about 10 miles, the diameter of the cast-iron pipe 
being very nearly 1 ft. (exactly 300 mm.), Professor Unwin 
deduced for the coefficient /, in this particular case, the value of 
0.0029.* It must be remembered that this is not the coefficient 
for a straight piece of cast-iron piping; including as it does, 
bends and joints, and also a small amount of leakage. Though 
transmissions to such distances are unusual, the value just quoted 
for the coefficient is probably a good average value to use for a 
practical case for the same diameter of piping. That is, in 
any practical case, for piping of an equal diameter, we should 
probably find the average losses per given length, approximately 
the same. 

From the conditions we have pi = 88.2 lbs. per square inch, 

%D 2 
T = 522, L = 26,400 ft., D = l ft., and §i = (1200/60)/ = 25.5 

ft. per second. For /, we will use the value 0.0029. Substituting 
these values, in equation (71), we find 



QQ o/i 0-0029X25.5 X26,400 \l _ _ ., 
P2 = 88.2^1 4 29x52 2 ) =77.8 lbs. per sq. in. 

Thus giving a loss in pressure of about 11.8 per cent. It must, 
however, not be understood from this, that the percentage loss of 
power in transmission is also 11.8 per cent. The efficiency of 
transmission is found by taking the ratio of the work which the 
air motor can do in expanding adiabatically from the pressure 
P2 to that of the atmosphere, to that which would have been 
obtained had adiabatic expansion taken place before transmission. 
* "On the Development and Transmission of Power," by W. C. Unwin. 



COMPRESSED AIR AND COMPRESSORS 259 

That is, the efficiency of transmission is 

, .-M'-feTl -(ST _ 

,-v4-feH Htr 

Substituting in equation (72), for p a , pi, and p2, respectively, 
14.7, 88.2, and 77.8, we find, for the efficiency of transmission, 



2_ 
1 



/14.7 

= — \UA^_ 
u.7\t 



"°..2 

where n is assumed equal to 1.4. 

It is thus seen that, though the loss in pressure is about 
11.8 per cent, the loss in power, due to this loss in pressure, is 
only about 5.5 per cent. 

The efficiency of transmission may also be defined, depending 
upon the point of view, as the ratio of the work that could be 
realized, before transmission, by allowing the air to expand 
isothermally, to that which would be realized by means of isother- 
mal expansion after transmission. In any case, for pressures such 
as are ordinarily employed, the value found, for the efficiency of 
transmission, by this comparison will not differ materially from 
that found by means of equation (72) . If we make the computa- 
tion for this particular case, we find, by assuming isothermal 
processes, 

, 77.8 
l0g lA7 

*= -88<r - 930 ; 



log 



14.7 



which differs approximately, only 1.5 per cent from the value 
found by comparing adiabatic processes. 



260 



THERMODYNAMICS 



In the case of water, the coefficient /, other things being equal, 
is a constant for all diameters. This, however, is not the case 
for gases. In the case of air, the coefficient / is some function of 
the diameter. Various empirical formulae have been proposed, 
for cast-iron piping, by means of which / is found, in terms of the 
diameter. None of them, however, are true for all diameters. 
As an example, the following formula, proposed by Professor 
Unwin, may be cited. According to this formula, the coefficient 
is 



/= 0.0027 



b+m)- 



However, by computing the coefficient, for various diameters, 
by means of this formula, and comparing with the values, as found 
by actual experiments, it is found that there is considerable 
discrepancy, as the following table will show: 



D. 

In Feet. 


Coefficient. 


By Experiment. 


By Formula. 


0.492 
0.656 
0.980 


0.00449 
0.00377 
0.0029 


0.00435 
0.00393 
0.00351 



For the two smaller diameters there is very close agreement; 
but, in the case of the one of 0.98 ft. diameter, the discrepancy is 
considerable. Professor Unwin has proposed the value 0.003 for 
all diameters of 1 ft. or over. 

184. Composite Diagram. We are now prepared to show, 
by means of the p-v diagram, the losses for the compressor, the 
line, and the motor. Let, in Fig. 50, 01 and OH represent, respect- 
ively, the axes of zero pressure and zero volume; and the line 
AB the aspirating stroke. Assume further that the compressor 
is one working on two stages, compressing first adiabatically, 
in the low-pressure cylinder, from the pressure p a , as represented 



COMPRESSED AIR AND COMPRESSORS 



261 



by the point B, to a pressure p, as represented by the point C. 
At the point C, the exhaust-valve of the low-pressure cylinder 
opens and the air is expelled to the receiver at the constant pres- 
sure p. In the receiver, the temperature falls to its initial value, 
and he volume shrinks by an amount represented by CD; the 
point D being on the isotherm BF. The condition of the same 
mass of air now, as regards pressure and volume, at the end of 
the aspirating stroke, in the high-pressure cylinder, is represented 
by the point D. Compression now takes place adiabatically 
from the point D, to the point E, to a pressure pi. When the pres- 




sure pi has been attained, the exhaust- valve opens and expulsion 
takes place under constant pressure, as represented by the line 
EG. In the reservoir, the temperature of the air falls to its initial 
value, and the volume shrinks by an amount EF; the point F 
being on the isotherm BDF. During isothermal transmission, 
the pressure falls by an amount represented by GK; and at the 
end of admission in the air motor, i.e., at the point of cut-off, the 
condition of the air, as regards pressure and volume, is represented 
by the point L. The point L is again on the isotherm BDF. 
From the point of cut-off, L, the air expands adiabatically, as 
represented by the curve LM. At the point M release occurs; 
and the expulsion stroke is represented by the line MA . 

From an inspection of the figure it is evident that the net 
work done, per cycle, by the compressor, is measured by the area 



262 THERMODYNAMICS 

A BCD EG; and, the net work recovered by the air motor, per 
cycle, is measured by the area KLMA . Hence, the total loss of 
work is measured by the area BCDEGKLM. The work lost, 
per cycle, due to the compression and expansion not being isother- 
mal, i.e., in the compressor and motor, is measured by the area 
BCDEFLM; and, the work lost in transmission is measured by 
the area FGKL. In general, the thermodynamic loss in the com- 
pressor and motor is large, in comparison with the loss, due to 
friction, during transmission. 

185. Theoretical Efficiency of System. It will now prove 
instructive to assume a concrete case and make, without consider- 
ing other losses, a comparison between the three losses; that 
is, the thermodynamic loss, due to the compression, in the com- 
pressor, being adiabatic in place of isothermal, the loss in trans- 
mission, and the thermodynamic loss, due to the expansion in the 
motor, being adiabatic in place of isothermal. 

Let it be required to compress the air to a pressure of six atmos- 
pheres by means of a two-stage compressor. The work required, 
per pound of air, according to equation (23), will be 



w 2n 
n — r 



n-l 



= ^|x27,800(6 7 -l) = 56,800 ft.-lbs. per pound. 



If the expansion now take place isothermally, after cooling, the 
work recovered will be 

W 2 = p a Va log ^ = 27,800 log 6 = 49,800 ft.-lbs. per pound. 



This is a loss of about 12.3 per cent. 

Assume, now, that the transmission line has the same constants 
as that discussed in Art. 183. Then the pressure at the end of 



COMPRESSED AIR AND COMPRESSORS 263 

the line will be 77.8 lbs. per square inch; and the work that can 
now be recovered, due to isothermal expansion, will be 

F 3 = 27,800 log y~== 46,300 ft.-lbs. per pound. 

This is a loss of about 6.2 per cent of the total work. 

If the expansion now takes place adiabatically, the work done 
on the air motor is 

2_ 

TF 3 = 27,800X^|{l-(^) 7 | =36,900 ft.-lbs. per pound. 

This gives a loss in the air motor of about 16.6 per cent of the total 
work done. 

We have then, the following : Per Cent. 

Loss in compressor 12.3 

Loss in transmission 6.2 

Loss in motor 16.6 

Efficiency of system 64.9 

Total ....100 

From the foregoing computations, it is obvious that the 
efficiency of the system is low, not due to the loss in transmission; 
but on account of the combired losses in the compressor and 
motor. 

Assume, now, that the air is reheated to a temperature 275 °F. 
above the surroundings. The work which the air will now do 
on the motor is 

T 7Q7 

W^=~Ws=^X 36,900 = 56,300 ft.-lbs. per pound; 
* 1 i ozz 

where T is the temperature to which the air is heated before 
being admitted to the motor. This gives, for the gain in work, 
for the same quantity of air consumed, by the motor, approxi- 
mately 52.6 per cent. 

To make a comparison now, between the work done on the 
motor and that done on the compressor, it will be necessary to 



264 THERMODYNAMICS 

add to the work done on the compressor, the work due to the heat 
consumed in reheating the air. The heat consumed in elevating 
the temperature of 1 pound of air from the temperature T\ to 
the temperature T } is 

Q = C V {T-T 1 ) B.T.U. 
And the work which would be realized on a Carnot cycle, is 

Substituting the various values, we find 

W 5 = 778 X 0.238 X 275 X ^y = 17,600 f t.-lbs. 

Taking the ratio now, of W± to the sum of W\ and TF5, we find, 
for the efficiency of the system, 75.7 per cent, as against 64.9 per 
cent, obtained without reheating. Commercially, however, the 
gain is greater than that indicated by the computations. For, 
as previously stated, a low-grade fuel may be employed, and the 
motor operates better, especially so if a small percentage of water 
is injected into the heater. This water is evaporated in going 
through the heater, and condensed in going through the motor. 
There is involved in this operation a small thermodynamic loss; 
but otherwise, the effect is good, since the water present helps to 
prevent leakage. Finally, reheating has the effect of increasing 
the capacity of both the compressor and line. 

It must be emphasized that in no case are efficiencies obtained 
as high as those indicated by the foregoing computations. Due 
to imperfect valve action, leakage, and mechanical losses, in both 
the compressor and motor, the efficiency of the system may be 
reduced by 10 to 15 per cent below that indicated by the com- 
putations.* 

* For actual tests on air transmission systems, see Unwin, "On the 
Development and Transmission of Power." 



CHAPTER XVII 
REFRIGERATION 

186. The object of refrigeration is to maintain the temperature 
of some body, or aggregation of bodies, at some point lower 
than that of the surroundings. This may be done in two ways. 
One method is to abstract heat directly, by means of a refriger- 
ating machine, from the medium surrounding the bodies. The 
other method is to bring about the desired lowering of temperature 
by means of ice. The ice employed, to bring about the desired 
refrigeration, may be harvested, during the cold season, from 
rivers and lakes, or else, the so-called " artificial ice," produced 
by means of refrigerating machines, may be used. 

Since the putrefaction of organic growths, such as foodstuffs, 
is retarded with lowering of temperature, and, in general com- 
pletely prevented when the temperature becomes sufficiently 
low, the prime object of refrigeration is not the maintaining of 
low temperatures, but rather the effects due to such low temper- 
atures; i.e., the preservation of foodstuffs during storage and 
shipment, and, in general, the promotion of health and comfort. 

187. Commercial Refrigerating Machines. Refrigeration may 
be brought about in various ways. But commercially successful 
refrigerating machines are restricted to two types; viz, refriger- 
ating machines in which air is the working substance, and machines 
in which some volatile liquid, such as ammonia, or carbon-dioxide, 
is employed as a working substance. For ammonia machines, 
there are again two distinct methods of operation; viz, com- 
pressor machines, and absorption machines. These various types 
will be discussed subsequently under separate headings. 

265 



266 THERMODYNAMICS 

All commercial refrigerating machines operate as reversed 
engines ; but, it must not be understood from this that the machine 
is reversible. The working substance abstracts heat from a 
body of, relatively, low temperature, called the refrigerator, con- 
sumes energy either in the form of mechanical work or heat, 
and rejects heat to a condenser or cooler. The heat rejected to 
the cooler, barring various losses, is equal to the heat taken from 
the refrigerator plus the heat equivalent of the energy consumed 
in bringing about the transfer. 

Equation (24), of Chapter VIII, states that, for an engine 
operating reversed, on a Carnot cycle, 

W = JH 2 ^; (1) 

where W is the energy consumed in bringing about the transfer, 
H2 the heat abstracted from the refrigerator, S the temperature 
of the source, and R the temperature of the refrigerator. The 
source, in the case of a reversible engine, corresponds to the cooler 
of a refrigerating machine. 

In discussing the Carnot cycle, it was found that, other things 
being equal, the greater the range in temperature, the greater the 
amount of work realized for a given quantity of heat abstracted 
from the source. On the other hand, equation (1) clearly indi- 
cates that, other things being equal, for a given quantity of heat 
H2, abstracted from the refrigerator, the wo k which must be 
done by the compressor decreases as the difference of temperature 
between the cooler and refrigerator is decreased. Hence, the range 
in temperature between refrigerator and cooler, for refrigerating 
machines, should be as small as possible. 

188. Air Refrigerating Machine. The air refrigerating system 
consists essentially of four parts; viz, a cold storage room, a 
compression cylinder, an expansion cylinder, and a cooler. The 
cycle is as follows : During the aspirating stroke, of the compressor 
piston, air flows into the cylinder, from the cold storage room, 



REFRIGERATION 267 

which during the return stroke is compressed, practically adia- 
batically, to the desired pressure, and expelled to the cooler. 
The cooler, usually, consists of a series of pipes in which the air is 
cooled by water circulating through the tank in which the pipes 
are placed. From the cooler the air passes into the expansion 
cylinder, where it does work on the piston, expanding practically 
adiabatically, and is finally exhausted, at a low temperature, to 
the cold storage room. The work done in the expansion cylinder 
is utilized in helping to drive the compressor. Hence the work, 
barring mechanical losses, which must be supplied to the com- 
pressor by means of some motor, is the difference between that 
done in the compression cylinder and that done in the expansion 
cylinder. 

189. Ideal Coefficient of Performance. To make a mathe- 
matical discussion, of the cycle just described, it will be necessary 
to assume ideal conditions. Let T 2 be the temperature of the 
air entering the cooler, at the end of the adiabatic compression, 
T a its temperature as it leaves the cooler and is admitted to the 
expansion cylinder, To its temperature at the end of the adiabatic 
expansion as it enters the cold storage room, and T± its temper- 
ature as it leaves the cold storage room and enters the compressor. 
It will now be assumed that the pressures in both the cooling pipes 
and cold storage room are constant throughout the cycle, and the 
machine is mechanically perfect. 

By equation (15), Art. 177, we have for the work done per 
pound of air, on the piston of the compressor, 

Wi=JC v {T 2 -T{). . . . , . (2) 

By equation (48), Art. 181, we have for the work done per pound 
of air, on the piston, in the expansion cylinder, 

W 2 = JC p (Ta-T ) (3) 



268 THERMODYNAMICS 

The work which must be supplied, to make the process take place, 
is the difference between W\ and W 2 ; i.e., 

Wz=JCA(T 2 -T 1 )-(Ta-T )\ (4) 

Since, according to the assumed conditions, the ratio of the ranges 
in pressures for the two cylinders are the same, we find 

T~T 2 (5) 

Substituting the value of To, as given by equation (5) in equation 
(4) we find 

W3 = JC P (T 2 -T 1 ) 7 ^~ a (6) 

The heat per pound of air, expressed in mechanical units, taken 
from the refrigerator, is 

W^JC^Ti-To). 

Substituting again, for To, its value, we find 

Wi-JcJ^Ti-T.) (7) 

Taking the ratio of TP4 to W3 we find, for the ideal coefficient of 
performance, 

= W 4= Ti 
73 W 3 T 2 -T x * ' ' w 

Equation (8) again shows that, the smaller the difference in 
temperature between refrigerator and cooler, the larger will 
become the ratio of the work equivalent of the heat abstracted 
from the refrigerator, to the work supplied. It is, of course, 
obvious that, due to the fact that it is practically uneconomical 
to construct cooling pipes of sufficient volume, such that the pres- 
sure throughout the cycle in the cooler is constant, and further, 
since the pressure in the cold storage room varies, the ratio, 



REFRIGERATION 269 

as expressed by equation (8), cannot be realized in practice. 
Furthermore, due to various losses, which must be experienced, 
in the case of an actual refrigerating machine, this ratio is still 
further reduced. Solving equation (8), for the work that must 
be supplied to a perfect machine, we find 

Wz = W4 J ^± (9) 

1 i 

The commercial efficiency of a refrigerating machine may be 
defined as the ratio of the work which would have to be done, 
for the given range of temperature and given quantity of heat 
removed from the refrigerator, on a perfect machine, to that 
actually required. If W*> is the work actually required, then the 
commercial efficiency is 

W 5 W 5 X Ti UU; 

The heat which must be carried away by the circulating water, 
in the cooler, per pound of air, is 

Hi = C v (T 2 -T a ) (11) 

The cycle of an air refrigerating machine may be conveniently 
represented by means of the T-y diagram. By equation (12), 
Art. 169, the change in entropy, when both the pressure and volume 
vary, is 

n -n = CAog^+C P \og v ^ (12) 

Pi Vi 

In the cycle just discussed it was assumed that the pressure, 
during the absorption and rejection of heat, remains constant. 
Hence, equation (12) becomes 

?2-9i = C,logg = C,log^; .... (13) 

and the heating and cooling curves, on the T-<p diagram, are loga- 
rithmic. Let, in Fig. 51, the point A represent the condition of 



270 



THERMODYNAMICS 



the air, as regards temperature and entropy, at the instant when 
it enters the compressor at the temperature T\. During the 
adiabatic compression the entropy remains constant and the tem- 
perature changes from T\ to T2, as represented by the line AB. 
The cooling then takes place, as represented by the curve BC, 
at constant pressure, to the temperature T a . The adiabatic 
expansion, from the temperature T a to the temperature To, is 
represented by the constant entropy line CD. Finally, the rise 
in temperature, in the refrigerator, at constant pressure, from To 







> 


B 


c^/ 








T a 


T L 


A 


D 


To^^ 






E 






F 



Fig. 51. 



to Ti, is represented by the curve DA ; and the cycle is completed. 
The heat abstracted from the refrigerator is measured by the area 
FADE, the heat rejected to the cooler is measured by the area 
FBCE, and the work done, on the compressor, is measured by the 
area A BCD. Finally, the ideal coefficient of performance is 
given by 

_ Area FADE 
V Area ABCD' 

Since the rejection of heat to the cooler, and the abstraction 
of heat from the cold storage room, both take place at constant 



REFRIGERATION 271 

pressure, equation (8) may be deduced in a very simple manner. 
If the heat abstracted from the cold storage room, for a given 
interval of time, is 

H 2 = C P (T 1 -T ), 

then the heat rejected to the cooler for the same interval of time, is 

Hi = Cp(T2 — To) 
Therefore, the ideal coefficient of performance is 

H 2 C^Tx-To) T x 



T} = 



Hx-H 2 C^-Taj-CviTt-To) T 2 -7Y 



which is the same as previously found. 

It must be emphasized that the equations deduced, in this 
article, do not represent conditions as found in actual practice. 
For the pressure, in the cooling pipes, of any actual refrigerating 
machine will vary considerably throughout the cycle. Hence, 
the actual coefficient of performance, even when all other losses 
are neglected, will be less than that indicated by equation (8) . 

Due to the fact that air has a low thermal capacity, air refrig- 
erating machines are necessarily bulky, and therefore, commer- 
cially uneconomical. However, there are certain places, as for 
example on board of ships, where it is inadvisable to use machines 
employing a volatile liquid. For, in the first place, there are possi- 
ble dangers from injurious escaping gases. But, even if the escap- 
ing gas is not injurious, there is always the possibility of a large 
leak, and consequently a total loss of the working substance, which 
cannot be replaced until the end of the trip. This, however, 
means a complete disablement of the plant. Hence, air machines 
are used only as a matter of expedience and not economy, in place 
of refrigerating machines employing a volatile liquid as a working 
substance. 

190. Compression Machines Using Volatile Liquids. Com- 
pression refrigerating machines, using a volatile liquid for the 



272 



THERMODYNAMICS 



working substance, consist essentially of the parts as represented 
diagrammatically, in Fig. 52. A is the compression cylinder 
where the vapor is compressed, and then expelled into coils 
immersed in water in B; B being the condenser, or cooler. If 
the vapor is just saturated as it leaves the refrigerating coils, 
superheating may take place, during compression; this however 
is usually very small in comparison with the heat of condensation. 
Due to the high pressure in the condenser, and the low temper- 
ature, maintained by the circulating water, the vapor condenses, 
gives up the superheat and heat of condensation, which is carried 
away by the water, and the liquid flows into C, the storage tank. 




Fig. 52. 



In the tank C, the liquid is under a pressure corresponding to that 
of its vapor, for the existing temperature; the temperature of 
the liquid in the storage tank, usually does not differ materially 
from that of the surroundings. As an example, if the liquid 
employed be ammonia and the temperature in the tank is 75°F., 
then the pressure of the vapor is approximately 140 lbs. per square 
inch. Due to this high pressure, under which the liquid is in C, 
it flows, through the expansion valve D, into the coils in the refrig- 
erator E. The pressure in the coils, due to the aspirating action of 
the compressor, is low. By regulating the expansion valve, or 
the speed of the compressor, or both, the pressure in the refriger- 
ator coils may be varied at pleasure. Since, when the liquid passes 
through the expansion valve, the process is adiabatic, and no work 



REFRIGERATION 273 

is being done, the total heat content remains the same. There- 
fore, for thermal equilibrium to obtain, when the pressure falls 
fcom.pi, that existing in the storage tank, to p 2 , that existing in 
the refrigerating coils, there must take place a certain amount 
of evaporation, such that 

h=h 2 +qr 2 ; (14) 

where hi and h 2 , respectively, are the heats of the liquid corre- 
sponding to the pressures pi and p 2 , q the amount of dryness, and 
r 2 the heat of vaporization at the pressure p 2 . From equation 
(14), we find 

hi — h 2 /1C * 

q= ~^ ; (15) 

and the remainder of the liquid can, then, if completely vapor- 
ized, take from the surrounding medium the quantity of heat 

H 2 = (l-q)r 2 (16) 

In order that heat may flow from the medium in E, into the 
coils it is, of course, necessary that the temperature of the medium 
be higher than that of the liquid, in the coils, corresponding to the 
pressure p 2 . If the difference of temperature is sufficient, the liquid 
will be completely vaporized; and the quantity of heat, as 
expressed by equation (16), will be removed from the refrigerator. 
If the difference of temperature be greater than this, the vapor 
becomes superheated ; and the quantity of heat removed from the 
refrigerator will be greater than that indicated by equation (16). 
The ideal p-v diagram, of the cycle just discussed, is represented 
in Fig. 53. The point A represents the condition, as regards 
pressure and volume, of the vapor at the beginning of the aspira- 
ting stroke, and the point B represents the condition at the end 
of the aspirating stroke; the line AB, therefore, represents the 
volume, due to complete vaporization under constant pressure. 
The curve BC represents the compression, which is nearly adia- 



274 



THERMODYNAMICS 



batic, CD represents the expulsion, and also the condensation, 
under constant pressure, in the condenser, and DE the change in 
pressure, and consequent change in volume, due to partial 
evaporation in passing through the expansion-valve. Therefore, 
the net work done, during the cycle, • is measured by the area 
A BCD. Fina ly the ideal coefficient of performance is given by 
the ratio of the work equivalent of the heat removed from the 
refrigerator to the work equivalent of the area A BCD. 

The refrigerating coils, in which the vaporization takes place, 
may be placed directly in a cold storage room, in the form of pipes, 




or else placed in a tank containing a solution of some salt, called 
brine. The freezing point for the brine must, of course, be lower 
than the temperature in the coils. The brine may then be em- 
ployed, by circulating it through pipes, to bring about refrigeration 
in some place remote from the plant, or else, to produce ice, by 
abstracting heat from water, contained in tanks, immersed in 
the brine. 

The cycle of a compressor refrigerating plant, using a volatile 
liquid as a working substance, is most instructively represented 
by the T— 9 diagram. However, before plotting the T— 9 dia- 
gram, it will be necessary to deduce an expression for the change 



REFRIGERATION 275 

in entropy, for the substance, when passing through the expan- 
sion valve. To do this, let T\ be the temperature of the liquid 
in the storage tank, then by assuming some arbitrary temperature, 
say To, from which the entropy is measured, the entropy of a 
unit mass of the liquid, before passing through the expansion 
valve, is 

C Tx dT . Ti 
n = c\ -Y=c\og^; (17) 

where c is the thermal capacity, of the liquid, per unit mass. 
Assume some temperature T, in the refrigerating coils; T being, 
of course, less than T\. The entropy, then of a unit mass of 
liquid and vapor, measured from the same zero, is 

'T 



C T dT ar 
* 2 = C ) T -T+T 



Tar. 



- clog^+l^; .... (18) 

where q is the amount of dryness, and r the heat of vaporization 
corresponding to the temperature T. Taking the difference 
between equations (18) and (17), we find, for the change in entropy 
in passing through the expansion valve, 

T Ti or 

9 = <p 2 - n = c log^r -clog y o + t' ' ' ' ^ 

But, 

qr = h 1 -h = c(T 1 -T); 

substituting this value of qr in equation (19), and simplifying, 
we find 



A T .T x 



i; (20) 

Differentiating equation (20) with respect to T, Ti being assumed 
constant, we find 

S=-l(H w 

Equation (21) shows, since Ti>T, that as the temperature 
increases, the entropy decreases, and vice versa. Hence, the 



276 



THERMODYNAMICS 



entropy of the substance is increased by passing through the 
expansion valve. This is necessarily so, since the process is irre- 
versible. 

The T- <p diagram, Fig. 54, indicates the various parts of the 
cycle. BC is the constant entropy line for the adiabatic com- 
pression of the vapor, from the temperature T 2 to T s ; if the vapor 
be just saturated, as shown, when the compression begins, it will 
become superheated during compression. CK represents the 
cooling of the vapor to the temperature of condensation, T\\ 







7 


C 


D 


Ti 


k/ 




J 








/ 








/ 








/ 








/ 








/ I 








/ 




\ 




a/ V 


T 2 


\ 


B 


F| G| 




H 





Fig. 54. 



and KD represents the condensation of the vapor, in the con- 
denser, at the constant temperature T\. DA represents the tem- 
perature entropy curve for the cooling of the liquid, without 
expansion, from the temperature T\ to T 2 . Had evaporation 
taken place, without expansion, after cooling along the curve DA, 
the quantity of heat removed from the refrigerator would be meas- 
ured by the area FABH. Due to expansion, however, through 
the expansion valve, the entropy of the substance increases in 
changing from the temperature Ti to T2, as indicated by the curve 
DE. The curve DE is determined by assuming various values 
of temperature, between T\ and T 2 , and solving, by means of equa- 



REFRIGERATION 277 

tion (20), for the corresponding entropy. Hence the heat that it 
is now possible to remove from the refrigerator, in bringing about 
complete vaporization, is measured by the area GEBH. Con- 
sequently, the amount of refrigeration that is lost, due to the 
change in entropy, in passing through the expansion valve, is 
measured by the area FAEG. Had there been superheating in 
the refrigerating coils, the quantity of heat removed from the 
refrigerator would be increased; but, due to this superheating, 
the vapor at the end of the compression will, likewise, be super- 
heated by an additional amount. 

Since the evaporation, which takes place while the liquid 
passes through the expansion valve, has no refrigerating value, 
but merely brings about thermal equilibrium, by reducing the 
temperature of the liquid to that existing in the coil, it follows 
that the change in entropy, along the curve DE, depends upon 
the ratio of the heat of vaporization to the thermal capacity of 
the liquid. The higher the ratio of the heat of vaporization to 
thermal capacity of liquid, the smaller the amount of vaporization 
required, for a given difference of temperature, to reduce the tem- 
perature of the liquid to that existing in the coil; and consequently 
the smaller will be the area FAEG. Therefore, since the work 
done by the compressor is independent of the amount of vapor- 
ization that takes place, along the curve DE, it follows that a 
liquid for which the ratio, of heat of vaporization to thermal 
capacity, is high, is best suited, from an economical standpoint, 
for refrigerating purposes. 

191. Absorption Machines. The principle of operation of an 
absorption refrigerating machine is based on the fact that the volume 
of ammonia vapor that can be absorbed by a given volume of 
water, other things being equal, depends upon the temperature, 
and decreases rapidly as the temperature is increased. Hence 
if water, at a low temperature, is saturated with ammonia vapor, 
then, to drive off the vapor, heat must be absorbed by the water. 
Likewise, if ammonia vapor be passed into water at a low tern- 



278 



THERMODYNAMICS 



perature, absorption will take place with a consequent develop- 
ment of heat. 

An absorption refrigerating machine is represented, diagram- 
matically, in Fig. 55. A is a storage tank containing ammonia 
from which expansion takes place through the valve B, into 
refrigerating coils in C, where refrigeration takes place. From 
the coils in C, the ammonia vapor passes into the liquid in the 
absorber D. The liquid in D is a solution of ammonia in water, 
of slight concentration and, relatively, low temperature. The, 




Fig. 55. 

liquid in the absorber being at a low temperature and only slightly 
concentrated, the incoming vapor is readily absorbed. The liquid 
of high concentration is removed from the bottom of D, by means of 
the pump P, and forced, at a, into the generator F. In the gener- 
ator is placed a heating coil H, by means of which heat is supplied 
to the highly concentrated solution, and raises its temperature. 
Due to the high temperature, part of the vapor is expelled from the 
solution, under a high pressure, and passes into the condenser G. 
The condenser is maintained at a, relatively, low temperature by 
means of circulating water. Due to this low temperature and the 
high pressure, the vapor condenses and flows into the storage 
tank A. By means of the valve I, the pressure in G and A is regu- 



REFRIGERATION 



279 



lated. Finally, the solution of low concentration, at the bottom 
of the generator F, is forced, due to the high pressure subsisting 
in the generator, into the absorber, at b. The pipes which convey 
the highly concentrated solution into the generator at a, and the 
solution of low concentration into the absorber at b, both pass 
through the heat exchanger E. In the heat exchanger the solution 
at a low temperature, going from the absorber to the generator, 
takes up heat from the high temperature solution, going from the 
generator to the absorber. 

The cycle is, then, as follows : The absorption, in the absorber, 
corresponds to the aspirating stroke of the compressor, as repre- 




Fig. 56. 

sented by AB of Fig. 56. The change in pressure in going from 
the absorber through the generator is represented by the curve 
BC, and corresponds to the compression curve of the compressor. 
The line CD represents the condensation at constant pressure, 
the same as in Fig. 53. Finally, the curve DE represents the fall 
in pressure and consequent increment in volume, due to partial 
evaporation of the liquid, in passing through the expansion- 
valve B. There is, of course, in this cycle, as well as in the com- 
pressor cycle, due to a partial evaporation when the liquid passes 
through the expansion-valve, a loss in refrigeration. Furthermore, 
there is, due to the fact that the ammonia vapor when distilled 
in the generator, carries with it a certain amount of aqueous 
vapor, an unavoidable loss. 



280 



THEKMODYNAMICS 



Thermodynamically speaking, the ideal coefficient of per- 
formance of the absorption machine, as well as that of any other 
refrigerating machine, is given by 

R 



S-R' 



where S is the temperature of the condenser, and R the tempera- 
ture in the refrigerator. On the other hand, the commercial 
efficiency is given by the ratio of the work which would have to 
be done on a perfect engine, to bring about the given transfer of 
heat, to the energy actually consumed. That is, if H2 is the quan- 
tity of heat abstracted from the refrigerator, during a given 
interval of time, Hi the quantity of heat supplied to the generator, 
and W the work done on the pump, during the same interval 
of time, the commercial efficiency is given by 



JH< 



Er = 



S-R 



JHx + W 



(22) 



The following table, taken from " Landolt and Bornstein," 
is given to show how the coefficient of absorption for ammonia 
vapor, under normal pressure, varies with the temperature: 



X 


K 


T 


K 





98.7 


15 


60.6 


1 


92.7 


16 


59.1 


2 


87.7 


17 


57.6 


3 


83.6 


18 


56.1 


4 


79.9 


19 


54.7 


5 


77.3 


20 


53.5 


6 


75.6 


21 


51.9 


7 


73.9 


22 


50.6 


8 


72.3 


23 


49.6 


9 


70.6 


24 


48.6 


10 


68.9 


25 


47.6 


11 


67.2 


26 


46.5 


12 


65.5 


27 


45.5 


13 


63.7 


28 


44.4 


14 


62.1 


29 


43.4 



REFRIGERATION 



281 



where t is the temperature in degrees centigrade, and K is the 
number of grams of ammonia vapor absorbed per 100 c.c. of 
water. 

By heat of dilution of a substance is meant the quantity of 
heat which is evolved when a unit mass of the substance is diluted 
to an extent such that practically no more heat is evolved upon 
further d ut'on. 

According to experiments by Berthelot, when 1 gram of 
liquid ammonia has been dissolved in n grams of water, and this 
solution is then fully diluted, the heat evolved is as given in the 
following table : 



n 


Gram Calories. 


n 


Gram Calories. 


1.04 
1.06 
1.13 

1.98 


75.6 
74.4 
68.8 
40.0 


3.18 
3.76 
6.11 
10.1 


22.6 

18.8 ' 
12.3 
0.12 



The results given in the table were obtained from experiments 
conducted at temperatures of 14 °C. 

By inspection it is seen that very little heat is evolved after 
the dilution is greater than 10 to 1. It has been proposed to 
employ the empirical equation 



H = -, 

n 



(23) 



for the heat of dilution when a solution, of 1 gram of ammonia 
dissolved in n grams of water, is fully diluted; H being the heat 
evolved, and h some constant. If equation (23) be applied to 
the values as given in the table, the value 78 be assigned to h, 
and the values of H computed and compared with the observed 
values, an idea will be obtained as to how closely the empirical 
equation conforms to the actual experimental results. 



282 



THERMODYNAMICS 



n 


Observed. 


Computed. 


n 


Observed. 


Computed. 


1.04 


75.6 


75.0 


3.18 


22.6 


24.5 


1.06 


74.4 


73.6 


3.76 


18.8 


20.7 


1.13 


68.8 


69.0 


.6.11 


12.3 


12.8 


1.98 


40.0 


39.4 


10.1 


0.12 


7.7 



The foregoing table shows that, when the initial dilution is 
not greater than 6 to 1,, equation (23) gives fairly consistent 
results; but, for initial dilutions greater than this, the equation 
breaks down completely. Furthermore, since the lowest initial 
dilution in Berthelot's experiments was 1.04 to 1, equation (23) 
is necessarily doubtful for initial dilutions lower than this. 

Experiment shows that, if 1 gram of ammonia vapor is 
absorbed by water and completely diluted, there is evolved a quan- 
tity of heat equal to 496 gram calories; hence, if m grams of 
ammonia are absorbed, and complete dilution take place, there 
will be evolved 496m gram calories. Therefore, if we assume 
equation (23) to hold, there will be evolved, when m grams of 
ammonia vapor are absorbed by n grams of water, 



H! = mQ- 



mh 

n/m 



m 



(«- 



m. 



(24) 



where n/m is the number of grams of water per gram of ammonia, 
and Q the quantity of heat evolved when 1 gram of ammonia 
vapor is absorbed by water and completely diluted. If, now, 
m-\-k grams of ammonia be absorbed by n grams of water, the 
number of grams of water per gram of ammonia will be n/(m-\-k). 
Therefore, the quantity of heat 



H 2 =(m+k){Q 



(q-^xa) 



(25) 



will be evolved. Taking the difference between the right-hand 
members of equations (25) and (24), for the quantity of heat 



KEFEIGEEATION 283 

evolved, when a solution containing m grams of ammonia to n 
grams of water, absorbs k grams of ammonia, we find 

H = H 2 -H 1 = (m+k)(^Q- r! ^Xh)-m(Q-^h\ 

= k\Q~(2m+k)\. . . (26) 



Substituting for Q and h the numerical values, we obtain 
H=k 
Equation (27) may be reduced to English units as follows : 



78 
496 (2m + k) \ gram calories. . . (27) 

To 



# = /b{893-^(2m+£0 Ib.T.U (28) 

That is, equation (27) is the expression for the heat, in B.T.U., 
which is evolved when a solution containing m pounds of ammonia 
to n pounds of water, absorbs k pounds of ammonia vapor. 

As previously stated, the foregoing equations are empirical 
and are true only between certain limits of initial dilution; further- 
more, since the heat of absorption and dilution varies with the 
temperature, the results obtained by means of these equations 
are to some extent doubtful. The equations have been deduced 
merely to show the method of attack. 

Under ideal conditions the heat developed in the absorber is 
equal to that required in the generator; but, since the temperature 
of the generator must be higher than that of the absorber, it is 
impossible to utilize the heat developed in the absorber. Hence, 
to maintain the process, heat must be supplied, by means of some 
independent source, to the high-temperature generator, and heat 
must be abstracted from the low-temperature absorber. 

In the case of a compression machine the energy consumed 
varies directly as the difference of temperature between the con- 
denser and refrigerator. This, however, is not so in the case of 



284 THERMODYNAMICS 

an absorption machine; hence, for a wide range in temperature, 
the absorption machine is thermodynamic ally superior. A 
further advantage, which is mechanical, is that no compressor 
is required. In certain cases the heating in the generator is brought 
about by means of exhaust steam, from engines, which is again 
economical. Finally, the power consumed by the pump in an 
absorption machine is small in comparison with the other quan- 
tities involved.* 

192. Comparison of Air and Ammonia Machines. It was 
stated in Art. 189 that, due to the low thermal capacity of air, 
refrigerating machines employing air as a working substance are 
necessarily bulky. It is impossible to make a general comparison; 
but a rough estimate may be obtained by assuming a concrete 
case. Let it be assumed that the temperature of the refrigerator 
is 32 °F., and that the range in temperature of the air in passing 
through the refrigerator is 100°F. One pound of air will then 
remove, from the refrigerator, 

C P (Ti-To) =0.238X100 = 23.8 B.T.U.; 

and to do this, the compressor must take in 12.4 cu.ft. The volume, 
per B.T.U. removed from the refrigerator, then is 

12.4/23.8 = 0.521 cu.ft. 

Assume, now, an ammonia compression machine, with a temper- 
ature of 70°F. for the reservoir. The dryness after passing 
through the expansion valve will be 

* r 2 540 ' 

and the quantity of heat that can be removed, by complete 
vaporization taking place at 32°F., is 

(l-g)r 2 = 540(l-0.078)=498 B.T.U. per pound. 

* For a comprehensive discussion of absorption machines see "Modern 
Refrigerating Machinery" by Hans Lorenz. 
f From Peabody's Steam Tables. 



EEFRIGERATION 285 

The specific volume of ammonia vapor, at 32°F., is approximately 
4.74 cu.ft. per pound. Hence we find, for the volume, per B.T.U., 

4.74/498 = 0.00952. 

Taking the ratio of the volume for air, to that for ammonia, we 
find 

0.521/0.00952 = 54.7. 

This shows that for the assumed conditions, other things being 
equal, the bulk of the compression cylinder of an air refrigerating 
machine is very large in comparison with that of an ammonia 
machine; but, further than this, the air machine must also have 
an expansion cylinder. For lower temperatures in the refriger- 
ator, the ratio of the two volumes becomes somewhat less. Assume 
the temperature of the refrigerator 15°F., then the dryness, after 
passing through the expansion-valve, is 

61/554 = 0.110; 

and the quantity of heat that can be removed, by complete 
vaporization, is 

554(1-0.110) =493 B.T.U. per pound. 

The specific volume for the vapor of ammonia at 15°F. is 6.68 
cu.ft. per pound; hence, we find for the volume, per B.T.U., 

6.68/493 = 0.0135 cu.ft. 

The volume of air that the compressor must now take in, at the 
temperature of 15°F., is approximately 12 cu.ft. ; hence the volume 
of air per B.T.U. , is 

12/23.8 = 0.504 cu.ft. 

Taking the ratio of the volume for air, to that for ammonia, we 
find 

0.504/0.0135 = 37.3. 

The animonia machine is also superior from the thermody- 
namic standpoint. By considering the two cycles, it is obvious 



286 THERMODYNAMICS 

that the cycle of the ammonia refrigerating machine, approaches 
the Carnot cycle much more closely than does the cycle of an air 
machine. For, in the ammonia cycle, the greater part of the heat 
is abstracted and rejected, respectively, during vaporization and 
condensation; i.e., at constant temperature. On the other 
hand, in the case of the air cycle, both the abstraction and rejec- 
tion of heat take place with continuously varying temperature. 
The foregoing may be illustrated roughly as follows : As previously 
shown, the work done by a compressor per cycle, if the compres- 
sion is adiabatic, is 

Wi=JC v (T 2 -Ti) per pound. 

The heat, expressed in mechanical units, removed by 1 pound 
of ammonia, from the refrigerator, is 

W 2 = Jr(l-q); 

and the ideal coefficient of performance is 

. TTg = r(l-g) 

^ Wi c v {T 2 -T 1 y •..■••• ^ 

It was shown, in Art. 189, that the ideal coefficient of per- 
formance of an air refrigeration machine is 

V'= y^=Y{ ••••••• (30) 

In equations (29) and (30), T\ and T 2 are, respectively, the 
temperatures before and after adiabatic compression. If we 
assume, now, that the ranges in temperature for the two machines 
are equal, we find, for the ratio of the performance for the two 
processes, 

V r(l-ff) 



V C P T! 



(31) 



If we are dealing with ammonia, and conditions are such as 
ordinarily obtain in refrigerating plants, then, in equation (31), 
the numerator and T\ will be practically equal. But, C p for 



REFRIGERATION 287 

ammonia vapor is approximately 0.53; hence, the coefficient of 
performance for the ammonia machine, as expressed by equation 
(29), is approximately double that for the air machine, as ex- 
pressed by equation (30). 

From the foregoing discussion it is obvious that, due to its 
enormous bulk, and consequent mechanical losses, together with 
its thermodynamic inferiority, the air refrigerating machine is 
very uneconomical, both from the standpoint of first cost and 
operation. 

193. The Kelvin Heating Machine. Before leaving the sub- 
ject of refrigerating machines, it will be interesting to consider 
a heat engine running reversed as a warming machine. This was 
suggested as early as 1852 by Lord Kelvin. To illustrate this, 
let it be desired to maintain the temperature of a room higher 
than that of the surrounding atmosphere. This may be brought 
about by the direct application of heat, or else by a heat engine 
running reversed. Assume the heating to be brought about by 
an air refrigerating machine, such as discussed in Art. 188, then 
during the aspirating stroke a charge of air flows into the com- 
pression cylinder at a temperature T a . This charge is now com- 
pressed to a temperature T\ and expelled into pipes, placed in 
the room which it is desired to heat, where heat is abstracted. 
After cooling, the air does work in the expansion cylinder and is 
expelled to the atmosphere. For a reversible engine, the heat 
rejected to the room is equal to the heat taken in from the 
atmosphere plus the heat equivalent of the work done on the air. 
If Hi is the heat rejected to the room, and H a the heat taken in 
from the atmosphere, then 

Hi = H a +AW; 
and 

Ti-Ta 



W=JH V 



Ti 



If, now, Ti — Ta, the required range, be small, then the heat 
equivalent of W will be a small fractional part of Hi. 



288 THERMODYNAMICS 

To illustrate further: Assume a situation where it is impos- 
sible to obtain fuel of any kind, but that there is available energy 
in the form of an electric current. Heating may then be brought 
about in two ways. That is, heat may be developed by passing 
the current through a suitable resistance, or else, the energy may 
be consumed in driving an electric motor, which in turn drives 
some form of reversed heat engine. In either case, the energy 
spent per unit time, due to the current consumed, is given by the 
product of e.m.f. and current. To make a simple comparison 
it will be necessary to assume certain conditions. Let the tem- 
perature of the atmosphere be 0°F., and that required in the heat- 
ing coils, so as to maintain the room at a proper temperature, 
be 165°F. Now, to bring about equal heating effects, the heat 
dissipated per unit time must be the same in each case. Let H 
be the heat required per unit time, then 

AEh=H] (32) 

where E is the applied e.m.f. and I\ the current consumed, when 
the heating is brought about by means of resistances. Assume 
now, a perfect electric motor driving a perfect warming machine. 
The power consumed to bring about the same heating effects, 
for the given temperatures, is 

AEI 2 = H^~^=^H, (33) 

where 1 2 is the current consumed by the motor. Solving by 
means of equations (32) and (33), for 1 2, we find 

33/i 

2 125 ' 

showing that for a commercial efficiency even as low as 26.4 per 
cent, the warming machine is thermodynamically equal to the 
direct method. And for efficiencies higher than 26.4 per cent, 
the warming machine is thermodynamically superior. 



CHAPTER XVIII 
STEAM TURBINES 

194. The detailed descriptions of the various types of steam 
turbines and the attendant mathematical discussions require an 
extended treatise. For such a treatise the reader is referred to 
Dr. A. Stodola's classical work, " Die Dampf-turbinen." * No 
attempt will here be made to do anything further than lay down 
the most elementary principles, so as to enable the student to 
take up reading matter, on the subject, of an advanced nature. 

In steam turbines, as well as in water turbines, there are impulse 
turbines and reaction turbines. However, in the case of water 
wheels, the types most frequently used are single stage; i.e., one 
stationary part, which carries the guides, by means of which the 
water is given the proper direction before entering the wheel, and 
one rotating part, carrying a number of curved blades, by means 
of which the energy stored in the water due to pressure and velocity, 
is absorbed. On the other hand steam turbines must be multi- 
stage, i.e., consist of a number of fixed parts called guides, and a 
number of rotating wheels, called rotors; otherwise the speed 
would be impracticably high. Fig. 57 is a diagrammatic repre- 
sentation for two stages of a multi-stage turbine. 

195. Impact on Curved Surfaces. Before proceeding to make 
any mathematical discussions for steam turbines, it will be well 
to consider a few cases for a non-expansive fluid, such as water, 
impinging on curved surfaces. Let, in Fig. 58, abc represent the 
section of a curved blade, having impinging upon it a stream of 
water with a velocity, relative to the earth's surface, represented 

* Translated by L. C. Loewenstein. 

289 



290 



THERMODYNAMICS 



in magnitude and direction by ha. This velocity is briefly 
designated as absolute velocity. The line ad represents in mag- 
nitude and direction the velocity of the blade. Now, while a 



GUIDE 



ROTOR 



GUIDE 



ROTOR 



Fig. 57. 



particle of water starting from a moves to e, a distance equal to 
Vi, the tip of the blade a has suffered a displacement vi, as repre- 
sented by ad. Hence the velocity of the water V r , relative to 




Fig. 58. 

the blade, is given by de. Therefore, the absolute velocity at 
entrance is equal to the vector sum of the velocity of the tip of the 
blade, at entrance, and the relative velocity. 



STEAM TURBINES 291 

If the water is to glide onto the blade, so that there shall be 
no shock, the tangent, to the tip of the blade at a, must be parallel, 
as shown by ka, to the relative velocity V r . If, now, there is 
experienced no friction by the water as it glides along the blade, 
and there is no sudden change in direction, then the magnitude 
of V r will not change; and the water will leave the blade, parallel 
to the tip at exit, as shown by ci. If V2 is the velocity of the tip 
at exit, then the vector sum of V r and V2, gives for the absolute 
velocity at exit, V2, as shown by cj. We then have, respectively, 
for the triangles of velocities at entrance and exit, ade and cij. 
If, from a we draw ag equal and parallel to V2, and close the tri- 
angle by eg, we find V t , the vector difference between I r i and V2', 
i.e., the total change of absolute velocity. Resolving V t into two 
components, one normal to the motion of the blade and the other 
parallel to the motion, we find V e , represented by ef, the total 
change of absolute velocity in the direction of motion. V e may 
be called the effective component, since this is the one producing 
the motion. From the diagram, it is obvious that the absolute 
velocity at entrance cannot be parallel to the motion of the blade, 
but must make some angle with it; otherwise entrance into the 
channel, included between abc and a'b'c', cannot take place. 
Similarly, at exit, there must be a normal component to carry the 
water away, so as not to interfere with the following blade. Fur- 
thermore, it is essential that the two surfaces of the blade tips, 
both at entrance and exit, come to a point and have, practically, 
a common tangent, parallel to the relative velocity, so that the 
following blade, a'h'c' , may glide into the stream without shock. 
Unless this is so, there will be a loss in efficiency; for, whenever 
a stream of water impinges upon a surface with shock, there are 
developed eddy currents which, when subsiding, develop heat and 
the energy thus consumed is dissipated to the surroundings. 

To find the theoretical efficiency of a system of blades, as 
depicted in Fig. 58, it is only necessary to take the ratio of change 
in kinetic energy, in passing through the channel, to the kinetic 



292 THERMODYNAMICS 

energy at entrance. Let M be the mass of water, per unit time, 

passing through the channel; thenthe kinetic energy, at entrance, 

is 

MVi 2 
Wi = ^- (1) 



The kinetic energy at exit is 



MV 2 



W 2 = ^-; (2) 



and the energy given up to the system is 



W 1 -W 2 = Ws = ~(V 1 2 -V 2 2 ) (3) 



Hence, the efficiency is given by 



Wi = V J 2 -V 2 2 



VY 1 V 1 — V 2 //( n 

Y 3 = wI = -^7^ ( 4 ) 



Equation (4) shows that V 2 should be as small as possible; which 
means that it must be normal to the direction of motion of the 
blades and just sufficient to carry the required quantity of water 
away from the channel. 

We may consider this in another manner. Since force is 
numerically equal to rate of change of momentum, the effective 
force, in producing motion, is 

F=MV e ; (5) 

where V e is the change of absolute velocity in the direction of 
motion. And the power developed, since power is numerically 
equal to the product of force and speed, is 

,P 2 = MV e v i; (6) 

where P 2 is the power developed. Equation (6) assumes vi and 
v 2 numerically equal. The power of the stream before imping- 
ing, since M is the mass of water which passes through the channel 
per unit time, is 

Pi = -^-; (7) 



STEAM TUKBINES 



293 



where Pi is the power delivered by the stream, 
of P2 to Pi, we find for the efficiency, 



Pi 



2V e v 1 
7i 2 " 



Taking the ratio 



(8) 



Equation (8) again shows, that if Vi and v\ are fixed, the efficiency 
is a maximum when V e is a maximum; i.e., the velocity com- 
ponent normal to the direction of motion is as small as possible. 

196. The Pelton Cup. Pelton wheels may be taken as repre- 
senting impulse turbines in the case of hydraulic motors. One 
of the cups, as used in this type of wheel, is represented diagram- 
matically in Fig. 59. Vi is the absolute velocity of the entering 




Fig. 59. 



jet, and vi the velocity of the cup, whose section is represented by 
abc. The velocity of the water relative to the cup is given by 



7 r = 7i-i;i. 



(9) 



Since the direction of motion of the water at exit makes an angle 
6 with the direction of motion of the cup, the component of the 
velocity of the stream at exit, parallel to the motion of the cup, 
is given by 

V e =(Vi-vi) cos (10) 



294 THEKMODYNAMICS 

And the absolute velocity, parallel to the direction of motion of 
the cup, at exit, is 

V 2 ' = v 1 -{Vi-vi) cos (11) 

The change in absolute velocity in the direction of motion, there- 
fore, is 

V a = V 1 -V 2 , = V 1 -{v 1 -(V 1 -v 1 ) cos G) 

= (7i-t>i)(l+cose). . . (12) 

If M is the mass of water, per unit of time, impinging on the cup, 
then, the force moving the cup is numerically equal to the product 
of change in velocity and mass ; hence we find for the force acting, 

F = M(V 1 -v 1 )(l+cosQ) (13) 

Finally, since power is numerically equal to product of force and 
speed, we have, by multiplying both sides of equation (13) by vi, 
for the power developed by the cup, 

P 2 = Ft;i=Jlft;i(7i-!;i)(l+cose). .... (14) 

It is obvious, from Fig. 59, that the motion of the water cannot 
be completely reversed; i.e., the direction of motion of the water 
leaving the cup must be inclined to the direction of motion of the 
cup. For, otherwise, the stream at exit will interfere with the 
forward motion of the following cup. 

By assuming, in equation (14), the power and the velocity of 
the cup variable, and the other quantities constant, we find, by 
differentiating for a maximum, 

^- = M(Fi-2t;)(l+cos 0) = O; 
from which 

That is, for maximum power, the velocity of the cup must be one- 
half the velocity of the stream. 



STEAM TURBINES 295 

Substituting in equation (14), this value for vi, we find, for the 
maximum power developed by the cup, 

P2 = M^(Vi-^)(l+cos 6)=^p- 2 (l+cos 0). . (15) 

Since the energy of the impinging jet is 

the efficiency of the cup becomes, 

P 2 _Fi 2 (l+cos 0)_l+cos 6 • 

*=K~~ 27? - — 2 * ' * ' (16) 

We may deduce the expression for the efficiency of the cup by 
considering the kinetic energy at entrance and exit. By an inspec- 
tion of Fig. 59, it is obvious that the absolute velocity at exit is 
the vector sum of (Fi — vi) and v\. Designating the absolute 
velocity at exit by V2, we find 

V 2 z = (V 1 -v 1 ) 2 +v 1 2 -2(V 1 -v 1 )vi cos 6 
= Fi 2 -2(Fiz;i-z;i 2 )(l+cos 0). 

And the efficiency becomes, since it is given by the ratio of the 
kinetic energy absorbed by the cup, to the kinetic energy of the 
stream, 

2(Fiz;i-z;i 2 )(l+cos0) 
ri = yj . 

Assuming Vi and constant, and the velocity of the cup and the 
efficiency variable, the expression for the efficiency becomes, 



f] 



2(Fi?;-z; 2 )(l+cos0) 



Vi 



and solving this for a maximum, we find 

dri 2(Fi-2^)(l+cos0) 



dv Vi 2 

from which 

Vi 



= 0; 



296 



THERMODYNAMICS 



Substituting this value of v, in the equation for efficiency, we find 
for the maximum efficiency 

l+cos0 

which is identical with equation (16). 

Equation (16) shows that if the angle 6 equals zero, so that the 
motion of the water is completely reversed, the efficiency equals 
unity. But, this also means, and necessarily so, since for max- 
imum efficiency wi = Fi/2, that the water is brought to rest; and 
the kinetic energy possessed by it, before impact, is completely 
absorbed by the cup. However, as previously stated, it is 




Fig. 60. 



impossible to have the angle equal to zero, on account of inter- 
ference with the following cup. Assume the angle to vary 
between the extreme limits of 0° and 90°. Then, as just stated, 
for the angle = 0, the efficiency is unity; and for the angle = 90°, 
the efficiency becomes 1/2. When equals 90°, the result is equiv- 
alent to normal impact without shock, as represented in Fig. 60. 
If Vi is the absolute velocity before impact, then, since for maxi- 
mum efficiency the velocity of the cup equals Fi/2, the absolute 
velocity of the stream after impact, is equal to 7i/V2. There- 



STEAM TURBINES 297 

fore, since the efficiency is equal to the ratio of ihe kinetic energy 
absorbed by the cup to the kinetic energy of the stream, before 
impact, we find 

v?-vm _l (17) 

Hence, theoretically, the efficiency of an impulse turbine may vary 
between 50 and 100 per cent; depending upon the value of the 
angle 0. In practice it is attempted to have the angle just 
sufficiently large so that the stream, leaving the cup, does not 
interfere with the following cup. In properly designed impulse 
wheels the actual efficiency may be, considering all losses, as high 
as 90 per cent. The results deduced clearly indicate, that in any 
case, it is essential, if a high efficiency is to be realized, to reduce 
the absolute velocity of the impinging stream, in going through 
the turbine, as nearly as possible, to zero. And this, in the case 
of steam turbines, is just as necessary a prerequisite for high 
efficiencies, as it is in the case of water turbines. 

The question, why is it possible, in the case of hydraulic 
motors, to convert so large a fractional part of the theoretical 
energy, due to the difference in topographical level, into actual 
work, and in the case of heat motors, so small a fractional part 
of the energy, due to the difference in " temperature level," 
naturally suggests itself. The answer is obvious. Every heat 
motor must act periodically. Even though the identical working 
substance is not used in the succeeding cycle, the result is just the 
same. For, the condition of the working substance, for the best 
results, must be at the beginning of each cycle the same as it was 
at the end of the preceding cycle. This is equivalent to cyclic 
operation. In the case of the hydraulic motor, however, the 
cycle is only partially completed. That is, the water, after 
having performed work, in falling through a certain height, is 
restored to its original condition by the action of the sun, which 
completes the cycle automatically. In other words, the water 



298 



THERMODYNAMICS 



at the height Hi, of the headrace, falls to the height H2, of the 
tailrace, and performs, theoretically, the amount of work 

w(H 1 -H 2 ); 

where Hi is the height of the headrace, H2 the height of the tail- 
race, and w the weight of water. But, to complete the cycle, the 
water must again be raised from the level H2 to H\. This is 
done at the expense of the radiant heat from the sun, by means of 
which the water from streams, lakes, the oceans, etc., is evapo- 
rated and carried, by means of convection currents, to higher 
elevations, where condensation takes place, and the difference in 
elevation, H1—H2, is again established. 

197. Flow of Fluids in Pipes of Varying Section (De Laval 
Nozzle). The flow of a gas or vapor, under steady conditions, 




Fig. 61. 

through a pipe of varying cross-section, is very similar to the 
flow of a liquid under similar conditions ; but, the flow of a gas 
differs materially from that of a non-compressible liquid in two 
respects. That is, in general, the weight of the gas, or statical 
head, is negligibly small in comparison with the pressure head 
and velocity head; but, on the other hand, account must be taken 
of the expansion. 

Consider a pipe CC , such as is represented in Fig. 61, and 
assume a steady flow, i.e., the mass of gas entering the section at 
C, for a given interval of time, is equal to the mass leaving the 
section at C", during the same interval of time. And further, 
that the gas flows, without friction, in straight stream lines. 
Let m be the mass of gas that enters the channel at C, per unit 



STEAM TURBINES 299 

time, with a speed s a , and pressure p a , then an equal mass will 
leave the channel at C, during the same interval of time, with 
some speed s b , and pressure p b . Assume the pipe CC, to be divided 
into an indefinitely large number of sections, such that the thick- 
ness of each section is indefinitely small. Let the pressures on 
the left-hand side of the various sections, be, respectively, 

Pa, P2, P3, • • • Pn-1, Pn', 

and on the right-hand side of the sections, 
V2, P3, m • • • Pn, p&. 
Representing the respective cross-sectional areas by 

■A-a, A.2, -A-3, . . . A. n , A b , 

and the corresponding thicknesses of the elements by 
ds a , ds2, dss, . . . ds n , 

then the work done by the positive pressures, during the time that 
the displacement ds a takes place at C, and the displacement ds n 
takes place at C ', is 

W ' = PaA a ds a -\-p2A 2 dS2-\- • • • +Pn-lA n - 1 dSn-l+PnAndSn- ■ (18) 

Likewise, the work done by the back pressures is 

W"= —p2 A 2 ds2 — p3A3ds3— ••• —pnAndsn — pbAbdsb. . . (19) 

Taking the sum, of equations (18) and (19), we find, for the net 
work done, due to change in pressure, 

W^W'+W'^PaAadsa-ptA-tdsi,. . . . (20) 

Changing, in equation (20), the subscripts a and & to 1 and 2, 
we have 

Wi=piAidsi — p 2 A 2 ds 2 . ' (21) 



300 THEKMODYNAMICS 

If dsi is the distance passed through, at C, in the time dt, and 
ds2 the distance passed through at C in the same interval of time, 
then 

ds\=s\dt, 
and 

dS2 = S2dt', 

where si and S2 are, respectively, the initial and final speeds. 
Substituting, in equation (21), these values of dsi and ds2, we obtain 

Wi = (piA 1 s 1 -p 2 A 2 s 2 )dt (22) 

Since, m is the mass of gas, flowing per unit time, we have, for 

steady flow, 

_ Aisi A 2 s 2 
m— = ; 

Vi V2 

where v\ and V2 are, respectively, the volumes per unit mass of 
the gas corresponding to the pressures pi and P2. Substituting 
in equation (22), we obtain 

Wi — ijpivi^^v^mdt . (23) 

The change in kinetic energy is 

and expressed in engineer's units, this becomes 

W 2 = Sj ~^mdt . (24) 

The work due to expansion is, 

Ws = (mdt) ( 2 pdv; ...... (25) 

and for the assumed conditions, 

Tfi + T^2+Tf 3 = 0. 



STEAM TUKBINES. 301 

Hence, 

Sl 2 _ g 2 rv 2 

(piVi — v 2 V2)mdt-i s — mdl-\-{mdt) I pdv = 0; 

Zg Jv l 

from which 

s 2 2 — s i 2 C v * 

— o - = pivi-p2^2+ I pdv (26) 

But, 



ri>2 rv\ 

p\Vi—p2V2 J r I pdv= I vdp; 

J v i J P 2 

2g--k ^ ■■■■■■ W 



hence, 

S2 2_. Ql 2 rVi 



If, now, the process be such that the equation 

pv n = k 
i_ _i_ 
holds, then v = k n p n ; and by substitution, equation (27) 

becomes 

s 2 2 — s i 2 — C Vx -— n — n ~ 1 VlzA 

___ = fc»J p ndp = ~_ I jkn(p 1 n -p 2 n ). . (28) 
1_ 

Substituting for & ra its value, we find 

s 2 2 -si 2 n L »=1 «=* w f /p 2 \^1 ™ 

___ = __ pi ^ 1 (p 1 B _ p2 n )= ___ Wl | 1 _^j , j (2g) 

Or, since 

L L L 

fcn z=p 1 n Z ; 1= p 2 n Z ; 2j 

1_ 

equation (28) may be simplified by dividing by k n , and multiply- 

i_ 
ing the first term in the parenthesis by pi n vi, and the second term 

i_ 
by p2 n V2. Performing this operation we find 

S2 2 — si 2 n , /o ^ x 

-^— = —^ViVi-p2V2). ..... (30) 



302 



THERMODYNAMICS 



If the pipe is curved, then the pressure on the convex surface 
is less than on the concave surface. However, unless the change 
in direction is considerable, the difference in pressure is very small. 

The foregoing conclusions are the result of a modification of 
of Bernouilli's theorem; i.e., applying the theorem to a compress- 
ible fluid of negligible weight. 

If the initial speed is negligible, as is the case when discharge 
takes place from a comparatively large vessel, then si, inequation 
(29), may be omitted, and we have 

n-l 



52 _ n 



PlVl 



1- 



(31) 



71-1 1 



Solving for S2, we find 

Now, the mass of fluid conveyed, per unit time, through any 
section is 

As 
m = — ; 



where A is the area of the section, s the speed, and v the volume 
per unit mass. But for steady flow this is a constant throughout 
the pipe. Hence we have 

As A2S2 






V2 



(33) 



where A 2 is the area corresponding to the pressure P2, and V2 
the corresponding volume per unit mass. By combining equa- 
tions (32) and (33) we find 

A2S2 (V2 



m = 



V2 



V2 [n — 1 



n-\ 



pm l 



(34) 



Assuming again, the flow to be such that the equation 



pv =pivi 



(35) 



STEAM TURBINES 303 

holds, we find 

Substituting this value of V2 in equation (34), we find 

-4ss(© ! -(gfif • • • <=>«» 

Since, as previously stated, m is constant, equation (34) may 
be written 

where A is any section, p the corresponding pressure, and v the 
corresponding volume per unit mass. Since As/v is a constant, 
s/t; must be a maximum, when A is a minimum. But, s/v becomes 
a maximum, and hence, A a minimum, when the value of the 
expression included in the bracket, of equation (37), becomes a 
maximum. Hence, we may write 



L =Kx= (W IP 



and 



from which 



Pi/ \Pi 



2-n 1 

dp n L n n + i > 

P\n pi n 



™fahl) ^ 

The value of p, as given by equation (38), is that value which 
makes s/v a maximum, and, therefore, A a minimum. Substitut- 
ing this value of p, for p2, in equation (32), and reducing, we find 

i^ 
/ 2gn \ 2 



•■^"i (39) 



304 THEKMODYNAMICS 

Finally, by means of equations (35), (38), and (39), we find 



i 
As A/2gn \2 

v v \n-\-r 



n+l vi\n+l, 



• (40) 



Equating the right-hand members of equations (36) and (40), 


we find 


A 2 
A 


2 

- n-1/ 2 \n-i - 
n+l\n+l/ 

2 » + l 

/pgV _ (v2\ n 

W W 


i 

2 

; . . . . . . (41) 



which gives the ratio of the area, for the pressure p2, to minimum 
area. 

Solving equation (38), on the assumption that the fluid is 
saturated steam, for which n equals 1.135, we find 



p = 0.577pi. 

By means of equation (32), the final speed may be determined, 
when the initial and final pressures are known; and with the aid 
of equation (37) any one of the three quantities, viz, m, A, and 
p, may be found, if two of them are given. 

Since, in deducing the foregoing equations, we equated work, 

expressed by the product of pressure and volume, against energy, 

expressed by the product of mass and square of the speed, we must 

in substituting numerical values, in these equations, use the same 

system of units. As an example, if in equation (39), s is to be 

given in feet per second, p\ must be given in lbs. per square 

foot, and v\ in cubic feet. Reducing equation (39), so that p\ 

is expressed in lbs. per square inch, and substituting for g and 

n the proper values, we find 

j_ 
/2X32.2X1.135 v/1 AA \2 __ ^ 

s=( 2^35 XUApiVi) =70.2(pit; 1 )l. 



STEAM TURBINES 305 

For steam under a pressure of 100 lbs. per square inch, the volume 
per pound is approximately 4.43 cu.ft.; hence, by substitution, 

we find 

s = 70.2(443) * = 1478 ft. per sec. 

If the pressure be 200 lbs. per square inch, for which the volume, 
per pound, is approximately 2.29 cu.ft., we find 

s = 70.2(458)* = 1503 ft. per sec. 

These two computations show that the variation in speed is small 
when compared with the variation in pressure. 

Reducing equation (40) in a similar manner, we find 

A f 2X32.2X1.135/ 2 XoIm Pl | 2" 

m = l44{ 2JL35 Ul35 ) X 144 ^} 

i_ 

= 0.30A f — J pounds per sec. ; 

where A is now in square inches. 

198. Two Principal Types of Turbines. From the previous 
discussions on the flow of steam through pipes it is obvious that 
the speed of flow, for any considerable difference in pressure, is 
very high; and that if any single-stage turbine, i.e., a turbine 
consisting of a set of nozzles and only one rotating part, were to 
utilize practically all the kinetic energy of the steam, due to its 
speed at nozzle exit, the speed of the turbine would have to be 
abnormally high. As an example, some of the De Laval tur- 
bines, which were single stage, had speeds as high as 40,000 r.p.m. 
Though the efficiencies of the De Laval turbines, from the stand- 
point of steam consumption, were not exceptionally low, the 
enormously high speeds were a serious disadvantage. For, in 
no other mechanical contrivance, not even dynamo electric 
machines, which are operated at relatively high speeds in com- 
parison with other machines, are such high speeds ever approached. 



306 THERMODYNAMICS 

Therefore, to utilize the power developed by a single-stage 
turbine it is necessary to employ a reduction gear. But, a reduc- 
tion gear means an additional first cost, and a lowering of mechan- 
ical efficiency. Furthermore, proper lubrication becomes exceed- 
ingly difficult when machines are operated under speeds such as 
are attained by single-stage turbines. 

The difficulties, however, stated in the preceding paragraph, 
were overcome by the introduction of multi-stage turbines* 
That is, by allowing the steam to act successively upon the rotors 
of a multi-stage turbine, its speed is gradually reduced, and the 
speed of the turbine need not be abnormally high. The turbine 
must of course be so designed that the steam expands, and the 
temperature is reduced continuously to the lowest possible value 
at exit. That is, the kinetic energy of the steam at exit must be, 
as nearly as possible, equal to zero. 

There is then the choice of the following types of multi-stage 
turbines: Combined impulse and reaction, and impulse. 

199. The Parsons Turbine. The Parsons turbine, at the pres- 
ent time, represents one of the commercially successful types of 
turbines; and may be considered a combined impulse and reaction 
turbine of the parallel-flow type. That is, the steam passes through 
the first set of guide blades approximately parallel to the shaft 
of the turbine, and has given to it the proper direction so that 
it may enter into the channels of the first rotor without shock. 
At exit from the first rotor, the steam enters a second set of guide 
blades, where it is again directed so as to properly enter the chan- 
nels of the second rotor, etc. In this way the steam reacts, 
expands, and falls in pressure continuously as it travels, from 

* There appears to be considerable confusion in regard to the meaning of 
the word " stage." In some cases authors designate a turbine, as an n-stage 
turbine when there are n rotors, which is consistent with the nomenclature 
employed in the case of hydraulic turbines. In other cases, however, namely 
the Curtis turbine, by number of stages is meant the number of sets of 
expanding nozzles. 



STEAM TURBINES 307 

entrance to exit, through the turbine. Since the steam is con- 
tinually expanding, the length of the blades and spacing, for both 
guides and rotors, must be increased so that the ratio of steam 
speed and blade speed, upon which the efficiency of the turbine 
depends, is maintained constant. 

200. The Curtis Turbine. The Curtis turbine is of the impulse 
type. The steam expands in a set of nozzles, where the pressure 
head is converted into velocity head, and then impinges on the 
curved blades of a rotor. Part of the kinetic energy of the steam 
is absorbed by the first rotor; the steam then at reduced speed 
passes through a set of guide blades where it is directed so as to 
properly enter the channels of a second rotor, where the speed 
is still further reduced, etc., until, finally, the speed is very low. 
The steam is then expanded through a second set of nozzles, and 
again passes through a series of rotors and guides, precisely as 
in the first stage. This is continued until the pressure of the 
steam has been reduced to the desired exhaust pressure. The 
number of stages, other things being equal, depends, of course, 
upon the range in pressure. Due to the fact that the speed of 
the steam is reduced in each rotor, the passages traversed by the 
steam must be continuously enlarged. This is brought about 
by reducing the curvature of the blades as well as lengthening 
them.* 

201. Comparison of Parsons and Curtis Turbines. Since 
the speed of the steam entering a Parsons turbine is moderately 
low, and for high efficiency its absolute velocity at exit must 
approach zero in value, it follows that the relative velocity must be 
high. That is, the relative velocity at exit being, approximately, 
the vector difference between the absolute velocity at entrance 
and the velocity of the wheel, it follows that, since the velocity 
of the steam at entrance is low, the relative velocity at exit will 
be high, and therefore, the velocity of the wheel must be high 

* For a comprehensive discussion on the design and testing of turbines 
see "The Marine Steam Turbine" by J. W. Sothern. 



308 THERMODYNAMICS 

in order that the absolute velocity at exit may be low. On the 
other hand, in the case of an impulse turbine, the velocity of the 
blade, for the best efficiency, is approximately one-half that of 
the entering steam. Hence it is obvious that, other things being 
equal, the Parsons turbine is inherently a higher speed prime 
mover than is the Curtis turbine. 

202. Turbines and Reciprocating Engines. No matter how 
operated the steam turbine is inherently a high-speed prime mover; 
and since power is proportional to the product of torque and angu- 
lar velocity, it follows that for equal output, the steam turbine, 
with its high rotative speed, will be of smaller dimensions than 
a reciprocating engine. Furthermore, where rotative motion is 
required, which is usually the case, the turbine needs no connect- 
ing rod and crank, as does the reciprocating engine. Again, 
where electric generators are direct connected, as in power plants, 
high speeds, up to a certain point, are desirable. Since, the power 
developed is equal to the product of e.m.f. and current, high-speed 
generators, for equal output, will have a lower first cost and occupy 
less floor space than low-speed generators. Finally, the turbine 
has the further mechanical advantage of having a uniform turn- 
ing moment. On the other hand there are certain cases where 
low speeds are essential either to successful operation or economy; 
under such conditions the reciprocting engine is superior. As 
an illustration of this we may consider present conditions in marine 
engineering. As previously stated, the turbine, using high pres- 
sure steam is, for high efficiencies, inherently a high-speed prime 
mover; on the other hand the propeller of a ship, is, for high 
efficiencies, inherently a low-speed mechanism. On passenger 
liners, the increased rates, which passengers are ready to pay for a 
reduction of time in transit, more than pay for the increased cost 
of operation. However, on freight steamers, such is by no 
means the case ; and it appears that for such steamers the recipro- 
cating engine combined with a low-pressure turbine, as regards 
economy, is at least equal if not superior to the turbine. 



STEAM TURBINES 



309 



Thermodynamically, the steam turbine is far superior to the 
reciprocating engine. For, in the turbine there is no alternate 
heating and cooling of the surfaces with which the steam comes 
into intimate contact. In other words, in the case of a turbine, 
very shortly after starting, steady conditions will prevail ; and the 
incoming steam, therefore, does not come into contact with sur- 
faces which have been previously chilled by the low-temperature 
exhaust steam. That is, the steam changes gradually in pressure 
and temperature from admission to exhaust. And this means that 




-G 



v 

Fig. 62. 

there is no condensation excepting that due to radiation. Hence, 
in a turbine, condensation is largely eliminated in comparison 
with a reciprocating engine; and herein lies one of the great 
factors that makes the turbine thermodynamically superior to 
the reciprocating engine. Another important factor is the fact 
that in a turbine a good vacuum is utilized to much better advan- 
tage. This is illustrated by Fig. 62. Let, in the figure, ABODE 
be the indicator diagram of a reciprocating engine operating 
between the pressures as indicated by the points A and E. Then 
the net work done by the engine is measured by the area ABODE', 
and the net work realized by means of a turbine, for the same 
limits in pressure, is measured by the area ABCFE. If, now, the 
back pressure be reduced, from that as represented by the line 



310 THERMODYNAMICS 

ED, to that represented by the line HI, the net gain in work, 
by means of a reciprocating engine, is measured by the area 
EDIH, and that, in the case of a turbine, by the area EFGH. 
That is, due to mechanical considerations, the length of stroke 
of the reciprocating engine is fixed; and hence, full expansion 
cannot be realized. But, in the turbine full expansion is realized 
and the toe of the indicator diagram is utilized in doing useful 
work. This gain in work, in the case of low-pressure turbines, 
is quite appreciable. 

203. Turbine Tests. It has been found impossible, up to 
the present time, to devise any method by means of which to 
determine the indicated power of a turbine, in the same manner 
that the indicated power of a reciprocating engine is determined. 
There is, however, no difficulty experienced in determining the 
output, or brake power. The output is determined in precisely 
the same manner as described in Art. 133; and the thermal effi- 
ciency is determined as described in Art. 136. If it be desired 
to determine the commercial efficiency, the method of procedure 
is precisely the same as that described in Art. 137. The com- 
parison frequently made is that between the actual output of the 
turbine, and that which would have been obtained on a Rankine 
cycle, as discussed in Art. 138. 

204. Reciprocating Engines and Low-pressure Turbines. In 
1906, H. G. Stott presented a paper* before the American Institute 
of Electrical Engineers on " Power Plant Economics," which gave 
a complete analysis of the various losses, from the coal bunkers 
to the bus-bars, of the Interborough Power Plant, located at 
Fifty-ninth Street and Eleventh Avenue, New York City. The 
prime movers employed at that time were of the Manhattan 
type compound Corliss engines; two engines being connected to 
one generator of 7500 K.W. maximum capacity. 

The following quotation is an extract from Mr. Stott's paper, 

* "Power Plant Economics," Transactions of the A.I.E.E., Vol. XXV. 



STEAM TUEBINES 311 

which is one of the most complete and instructive analyses that 
has ever been made of a power plant: 

" Three years ago the steam-power plant for the generation 
of electricity had apparently settled down to an almost uniform 
arrangement of standard apparatus in which one power plant 
differed from another only in details of construction of engines, 
generators, and auxiliaries. As only about twenty years had then 
elapsed since the first central station was put in operation on a 
commercial basis, this uniformity of design seemed to indicate 
that in the near future it would only be necessary to purchase a 
standard set of power-plant drawings, and make the necessary 
changes in size of units in order to have a station of the best type 
known to the art. 

" The internal combustion or gas engine had from time to 
time been brought forward as a candidate for the position of 
prime mover, with every prospect of improved economy in fuel 
consumption; but with the exception of a few special instances it 
was not looked upon with favor, as shown by the almost universal 
use of the steam engine. 

" After a long period of development a new factor in power- 
plant design; namely, the steam turbine, was placed on the market 
in commercial sizes. It is safe to say that during the last three 
years no other piece of apparatus has had so stimulating an effect 
upon the power plant. Its effect upon the entire plant has been 
most beneficial,- for it has revived the apparently moribund 
superheater. This has now been so developed and improved that 
superheat of 200° or 300° fahr. can be safely and economically 
obtained. With the development of the superheater further 
study of the problem of combustion has improved the efficiency 
of the furnace; and this most important subject is apparently 
susceptible of still further development. 

" One other important result of the steam-turbine develop- 
ment has been the development of condensing apparatus to such 
a point of efficiency that a vacuum within one inch of the simul- 



312 THEEMODYNAMICS 

taneous barometer reading can now be maintained without 
difficulty. 

" Another change in the power plant has been the reversion 
to high-speed generators, resulting in decreased cost of the gen- 
erator and its foundations, as well as saving in floor space. 

" Last but not least the steam turbine has put the recipro- 
cating engine and the gas engine on the defensive and has actually 
been unkind enough to throw out hints in regard to the applica- 
tion of Dr. Osier's proposed methods to the treatment of older 
apparatus. 

" The reciprocating engine and internal combustion engine 
have not been slow in accepting this challenge; they have 
responded by showing so improved an economy (especially in 
the gas engine) that the situation has become most interesting 
to the power-plant designer. It is safe to say that the develop- 
ments of the next ten years will show very marked improvement 
in power plant efficiency. 

" In regard to this development the author wishes to direct 
attention to the basic fact that in power plants one should not 
look merely for increased efficiency in the prime mover, but should 
also investigate and analyze the entire plant from the coal to the 
bus-bars: first, in regard to efficiency; secondly, in regard to the 
effect of load-factor upon investment; and thirdly, the effect of 
the first and second upon the total cost of producing the kilowatt- 
hour, which is the ultimate test of the skill of the designer and 
operator. 

" Efficiency. 

" In Table 1 will be found a complete analysis of the losses 
found in a year's operation of what is probably one of the most 
efficient plants in existence to-day and, therefore, typical of the 
present state of the art. 



STEAM TURBINES 



313 



" Table No. 1 

ANALYSIS OF THE AVERAGE LOSSES IN THE CONVERSION 
OF ONE POUND OF COAL INTO ELECTRICITY. 





B.T.U. 


Per Cent. 


B.T.U. 


Per Cent. 


1. B.T.U. per pound of coal supplied 

2. Loss in ashes 


14150 

441 
960 


100.0 

3.1 

6.8 


340 
3212 
1131 

28 

223 

203 

152 

51 

31 

111 

36 

28 

8524 

29 


2 4 


3. Loss to stack 


22 7 


4. Loss in boiler radiation and leakage . . 

5. Returned by feed-water heater 

6. Returned by economizer 


8.0 


7 Loss in pipe radiation 


2 


8. Delivered to circulator 


1 6 


9. Delivered to feed-pump 


1 4 


10. Loss in leakage and high-pressure drips 

11. Delivered to small auxiliaries 

12. Heating 


1.1 
0.4 

2 


13. Loss in engine friction 


8 


14. Electrical losses 


3 


15. Engine radiation losses 


2 


16. Rejected to condenser .... 


60 1 


17. To house auxiliaries 


2 








15551 
14099 


109.9 
99.6 


14099 


99.6 


Delivered to bus-bar 


1452 


10.3 





" Discussion of Data in Table 1 

" Item 1. B.t.u. per Pound of Coal Supplied. The thermal 
value of the coal used is evidently of prime importance, as it affects 
the cost efficiency of the entire plant. The method of purchasing 
coal used in the plant from which this heat balance is derived is 
that of paying for B.t.u. only, with suitable restrictions on the 
maximum permissible amount of volatile matter, ash, and sulphur. 

" A small sample of coal is automatically taken from each 
filling of the weighing hoppers, so that the final sample represents 
a true average of a boat-load of coal. This final average sample 
is then pulverized and tested for heat value in a bomb calorim- 
eter, after which a proximate analysis is made of another por- 



314 THERMODYNAMICS 

tion of the sample. This method of purchasing coal has been in 
use for two years, with highly satisfactory results. 

" Item 2. Loss in Ashes. It is doubtful whether a further 
saving in this item can be made, as the extra care and labor 
necessary to accomplish any improvement would in all probability 
offset the saving in coal. 

" Item 3. Loss to Stack. This is one of the most vulnerable 
points to attack, as the loss of 22.7 per cent, is very large. Recent 
investigations show that promising results may be obtained by 
the use of more scientific methods in the boiler room. In prac- 
tically all cases it will be found that this loss is due almost entirely 
to admitting too much air to the combustion chamber, result- 
ing in cooling of the furnace. This result is usually produced by 
" holes " in the fire; these " holes " may be due to several causes, 
but usually are due to carelessness on the part of the fireman. 

" Fortunately, a very valuable piece of apparatus has been 
placed upon the market in the shape of a CO2 recording instru- 
ment. The results of a series of tests made with this instrument 
are shown in Figs. 63 to 66. 

" Fig. 63 shows the average condition of a furnace using small 
sizes of anthracite, with forced draught. The conditions are such 
that approximately 40 per cent, of the thermal value is being lost. 

" Fig. 64 shows what improvement may easily be obtained by 
watching the CO2 record, and indicates a saving of about 19 per 
cent, over the previous case. 

" In the combustion of the small sizes of anthracite it is neces- 
sary to use a draught of not less than 1.5 in. of water; this breaks 
the crust of the fire in the thin spots, allowing the air to come 
through in such volumes that an enormous amount of heat is 
wasted in raising the temperature of the surplus air and at the same 
time causing inefficient combustion in the entire furnace. 

" Fig. 65 shows a record taken from a stoker boiler whilst the 
recorder was covered up to prevent the fireman from seeing the 
record. 



STEAM TUEBINES 



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318 THERMODYNAMICS 

" Fig. 66 shows a record taken from the same stoker boiler 
with the fireman watching the CO2 indications, resulting in a sav- 
ing of over 12 per cent. Later records show that even better 
results than an average of 11.4 per cent, of CO2 can be 
obtained. 

" Fig. 67 shows the calculated losses in fuel corresponding 
to various percentages of CO2 for three different temperatures of 
flue gases. 

" From a consideration of the above tests it seems reasonable 
to assume that the 22.7 per cent, loss to stack can, by scientific 
methods in the fireroom, be reduced to about 12.7 per cent, and 
possibly to 10 per cent. 

" Before the installation of the CO2 recorder a long series of 
evaporative tests was made to determine the most economical 
draught to carry when a high-grade semi-bituminous coal was 
burning on the automatic stokers. The results shown in Fig. 
68 were so remarkable that they were repeated under different 
conditions in order to confirm them. Since the installation of 
the CO2 recorder, however, the explanation is apparent; as the 
draught giving maximum evaporation per pound of combustible 
corresponds to the point of maximum CO2, illustrating the inher- 
ent difficulty of maintaining efficient conditions in the combus- 
tion chamber with high draught. This is well illustrated by Fig. 
69, showing the draught, per cent, of rating, and percentage 
of C0 2 . 

" Item 4- The loss in boiler radiation and leakage, amounting 
to 8 per cent., is largely due to the inefficient boiler setting of 
brick which, besides permitting radiation, admits a large amount 
of air by infiltration. This infiltration will increase with the 
draught, thus tending to exaggerate the maximum and minimum 
points on Fig. 68. The remedy for this radiation and infiltration 
loss is evidently to use new methods of boiler setting, such as an 
iron plate air-tight case enclosing a carbonate of magnesia lining 
outside the brickwork. 



STEAM TURBINES 



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THERMODYNAMICS 



























































































CURVES SHOWING THE RELATIVE FUEL LOSS 
TO PER CENT. C02 , BY VOLUME, IN FLUE GAS. 
DIFFERENCE BETWEEN TEMPERATURES OF 
ENTERING AIR AND EXHAUST GAS: 
CURVE "A"= 400° FAHR. 
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3IAin"10A -JLN30 UQd -00 



STEAM TURBINES 



321 



" Mr. W. H. Patchell,* of London, who recently visited us, 
has introduced very large boilers, assembling two in one setting; 
each boiler has a normal evaporation of 33,000 lb. per hour 
and in this way has cut down to a minimum the radiating surface 
per square foot of heating surface. He has also introduced the 
iron case with magnesia lining, and with good results. 

" The question of boiler leakage is one in which the choice 
of the lesser of two evils is necessary; for in the tubular or cylindri- 



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cal boiler the leakage will undoubtedly be less than in the water- 
tube type, owing to the smaller number of joints in the water 
space. But these two advantages are offset by the increased 
difficulty of construction, and the danger of using large boilers of 
the tubular type, especially with high-pressure steam. 

"It is now generally admitted that there can be no more 
difference in the efficiency of different types of boilers under 



* See paper read December 7, 1905, before the Institution of Electrical 
Engineers, by W. H. Patchell. 



322 



THERMODYNAMICS 



similar conditions than there can be in electric heaters, press 
agents to the contrary notwithstanding. 

" Item 5. Returned by Feed-water Heater. The importance of 
getting the feed water to the maximum temperature obtainable 
is generally recognized, and would seem to indicate that all auxili- 













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GRAPHICAL LOG OF AVERAGE READINGS 
TAKEN ON BOILER NO. 31, JAN. 1, 2, 1906. 


















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Fig. 



aries should be steam driven so that their exhaust may be utilized 
in the feed-water heater; in this way the auxiliaries may operate 
at about 80 per cent, thermal efficiency. 

" Item 6. Owing to the difficulty of pumping water at tem- 
peratures above 150 degrees fahr., when under pressure, it 
becomes necessary to install economizers for the purpose of 
increasing the feed- water temperature to 200 or 250 degrees fahr. 



STEAM TUEBINES 323 

As this increase of temperature is obtained from the waste gases 
at no expense for fuel, it only becomes necessary to consider the 
load-factor, as will be shown later, in order to decide whether 
economizers should be installed or not. In practically all cases 
where the load factor exceeds 25 per cent, the investment will be 
justified. 

' " In deciding upon the size of economizer to be installed it is 
important to consider first, the influence of the economizer upon the 
available draught due to the obstruction it offers and also due to 
the reduced stack temperature; the second important consider- 
ation is to equate the interest and depreciation charges against 
the saving in fuel, and so determine the amount of investment 
justified in each particular case. 

" Item 7. Loss in Pipe Radiation. By the use of two-layer 
pipe covering, each layer being approximately 1.5-in. thick, and 
sections put on in such manner that all joints are broken, the 
radiation losses have become practically negligible. 

" Items 8 and 9. Heat Delivered to Circulating and Boiler-Feed 
Pumps. As these auxiliaries may be either electrically driven or 
steam driven it is interesting to note that the thermal efficiency 
of the electrically-driven pumps would be equal to the thermal 
efficiency of the plant, multiplied by both the efficiency of con- 
version from the alternating to direct current and by the motor 
efficiency. In this case, there would be a net thermal efficiency 
of 10.3X0.93X0.90 = 8.63 per cent., whereas the thermal efficiency 
of the steam-driven auxiliary discharging its exhaust into a feed- 
water heater at atmospheric pressure would be approximately 
87 per cent. 

" Item 10. Loss in Leakage and High-Pressure Drips. The 
loss in leakage should be infinitesimal, and the high-pressure 
drips can be returned to the boilers, so that practically all the loss 
under this heading is recoverable. 

" Items 11, 12, and 17 are probably unavoidable and of so 
small a magnitude as not to merit much consideration. 



324 THERMODYNAMICS 

" Item 13. Loss in Engine Friction. Recent tests of a 7500- 
h.p. reciprocating engine show a mechanical efficiency of 93.65 
per cent, at full load, or an engine friction of 6.35 per cent. As 
this forms only 0.8 per cent, of the total thermal losses it is relatively 
unimportant. "Attention is called to the method of lubricating 
all the principal bearings by what is known as the flushing system, 
whereby a large quantity of oil is put through all the bearings 
by gravity feed from elevated oil reservoirs common to all the 
units; after passing through the bearings the oil is returned by 
gravity to oil filters in the basement and then pumped up to the 
reservoir tanks again. About 200 gallons per hour are put through 
each engine, and of this quantity only about 0.5 per cent, is lost. 
This method of oiling undoubtedly contributes to the general 
results. 

" Item 14- As large electrical generators can now be obtained 
which give from 98 to 98.5 per cent, efficiency, it would seem as 
if the limit in design had been reached and that hereafter the 
problem of design is to be merely one of altering dimensions to 
suit varying sizes and speeds. While this is true as far as the 
efficiency is concerned, other problems are continually arising, 
such as the design of generators for an overload capacity of 100 
per cent, to meet the demand for apparatus capable of taking care 
of great overloads economically for short periods, corresponding 
to peak loads of a railroad or lighting plant. 

" Item 15. Engine Radiation Losses. This source of loss has 
evidently been reduced to a negligible quantity by the use of 
improved material and methods of heat insulation. 

" Item 16. Rejected to Condenser, 60.1 per cent. This imme- 
diately introduces the thermodynamics of the steam engine, a 
subject so broad that it will be impossible to do more than touch 
upon some of the most important points in considering steam- 
engine efficiency. 

" The efficiency * of any heat engine can be expressed by the 
* Defined as ideal coefficient of conversion, Art. 89. — Author. 



STEAM TUEBINES 



325 



rp m 

ratio of E = — ^ where T\ is the absolute temperature of 

± 1 

the steam entering the engine and T2 the absolute temperature 

of the steam leaving the engine. Thus in the engine whose 

steam-consumption curve is given in Fig. 70, if the initial pressure 

is 175 lb. gauge and the vacuum at the low-pressure exhaust 

nozzle is 28 in., then the maximum thermal efficiency is — === — 

= 33 per cent. This would be true for any form of engine or 
turbine working between the same temperature limits. 



















































































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A. ECONOMY CURVE FOR 7 500 H.P. ENGINE LOAD 
EQUALLY DIVIDED BETWEEN CYLINDERS. 

B. ECONOMY CURVE FOR 7 500 H.P. ENGINE LOAD 
UNEQUALLY DIVIDED BETWEEN CYLINDERS. 










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4 000 5 000 6 000 

load: kilowatt- houp ( switchboard reading) 
Fig. 70. 



7 000 



" In Fig. 70, however, it is seen that the point of maximum 
economy shows a steam consumption of approximately 17 lb. 
per kilowatt-hour, which is equivalent to 20,349 B.t.u. per 
hour. One kilowatt-hour is equal to 3412 B.t.u. per hour, 
so that the actual efficiency of the steam engine and generator 
. 3412 



is 



20349 



16.7 per cent. As the generator efficiency at this 



326 THEKMODYNAMICS 

load is approximately 98 per cent, the net engine thermody- 
namic efficiency * is tt-^= 17 per cent. 
U.yo 

" The difference between the theoretical efficiency and the 
actual is then 33 — 17 = 16 per cent., of which 0.8 per cent, has 
already been accounted for in engine friction, so that the balance 
of 15.2 per cent, is due to cylinder condensation, incomplete 
expansion, and radiation. 

" As the engine friction in a two-bearing engine with high- 
pressure poppet valves and low-pressure Corliss valves has by care- 
ful design been reduced to less than 0.8 per cent, gain cannot be 
expected here, so attention must be centered on the loss due to 
cylinder condensation, etc., amounting to 15.2 per cent., in order 
to effect any improvement. 

" Superheated steam is the only remedy at hand and with it 
we can probably effect an improvement of 5 or 6 per cent, by using 
such a degree of superheat in the boilers that dry steam will be 
had at the point of cut-off in the low-pressure cylinder. 

" Any greater amount of superheat than this will merely 
result in loss to the condenser; for it should be remembered that 
the cylinder losses increase with the difference in temperature 
between the steam and exhaust portions of the cycle; in other 
words, the greater the thermal range of temperature the greater 
the condensation loss. This would seem to point to the use of 
more cylinders; but this involves additional first cost and fric- 
tion as well as more space and higher maintenance charges. 

" Fig. 71 shows what may be gained by reducing the temper- 
ature at the end of the cycle by means of increased vacuum, but 
in the case in point the maximum vacuum obtainable in practice 
was used so that no additional economy can be expected in this 
way. 

* Defined as thermal efficiency under Art. 136. — Author. 



STEAM TURBINES 



327 



" Summary of Analysis of Heat Balance. 

" The present type of power plant using reciprocating engines 
can be improved in efficiency as follows : 

Reduction of stack losses 12% 

Reduction in boiler radiation and leakage. . . . 5% 
Reduction in engine losses by the use of superheat 6% 



28 



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17 18 19 

water rate-pounds per kilowatt- hour-load 4 000 kilowatts 
Fig. 71. 



resulting in a net increase of thermal efficiency of the entire 
plant of 4.14 per cent., and bringing up the total thermal efficiency 
from 10.3 per cent, to 14.44 per cent." 

It was subsequently found that the capacity of the plant was 
inadequate to generate sufficient power to take care of the increas- 
ing demand, due to the increment in traffic; and it was finally 
decided to install low-pressure turbines to operate on the exhaust 
steam of the compound reciprocating engines. 



328 THERMODYNAMICS 

The following quotation * indicates the all-around gain by this 
combination. 

" During the year 1908 it became apparent that owing to the 
ever-increasing traffic in the New York subway, it would be 
necessary to have additional power available for the winter of 
1909-1910. 

" 2. The power plant of the Interborough Rapid Transit Com- 
pany, which supplies the subway, is located on the block bounded 
by 58th and_59th Streets, and by 11th and 12th Avenues, ad- 
jacent to the North River; it contains nine 7500-kw. (maximum 
rating) engine units, besides three 1250-kw. 60-cycle turbine units 
which are used exclusively for lighting and signal purposes. 

" 3. The 7500-kw. units consist of Manhattan-type com- 
pound Corliss engines, having two 42-in. horizontal high-pressure 
cylinders and two 86-in. vertical low-pressure cylinders. Each 
horizontal high-pressure cylinder and vertical low-pressure 
cylinder has its connecting rod attached to the same crank, so 
that the unit becomes a four-cylinder 60-in. stroke compound 
engine with an overhanging crank on each side of a 7500-kw. 
maximum rating 11,000-volt, three-phase, 25-cycle generator. 
The generator revolving field is built up of riveted steel plates of 
sufficient weight to act as a flywheel for the two engines con- 
nected to it. This arrangement gives a very compact two-bear- 
ing unit. The valve gear on the high-pressure cylinders is of the 
poppet type, and on the low-pressure of the Corliss double-ported 
type. 

" 4. The condensing apparatus consists of barometric con- 
densers, arranged so as to be directly attached to the low-pressure 
exhaust nozzles, with the usual compound displacement circu- 
lating pump and simple dry- vacuum pump. 

" 5. These engine and generator units are in general probably 
the most satisfactory large units ever built, as five years' experience 

* "Tests of a 15,000-KW. Steam-Engine-Turbine Unit," by H. G. 
Stott and R. J. S. Pigott. Transactions of the A.S.M.E., Vol. XXXII. 



STEAM TURBINES 329 

with them has proved; their normal economic rating is 5000 
kw., but they operate equally well (water rate excepted) on 
8000 kw. continuously. 

" 6. In considering the problem of how to get an additional 
supply of power, every available source was considered, but by 
a process of elimination only two distinct plans were left in the 
field. 

" 7. The electric transmission of power from a hydraulic 
plant was first considered, but owing to the high cost of a double 
transmission line from the nearest available water power, and the 
impossibility of gettng reliable service (that is, service having 
a maximum total interruption of not more than ten minutes per 
annum) from such a line, further consideration of this plan was 
abandoned. 

" 8. The gas engine, while offering the highest thermo-dynamic 
efficiency, at the same time required an investment of at least 
35 per cent more than ordinary steam-turbine plant with a prob- 
able maintenance and operation account of from four to ten times 
that of the steam turbine. 

" 9. The reciprocating-engine unit of the same type as those 
already installed, was rejected in spite of its most satisfactory 
performance, on account of the high first cost and small range of 
economical operation. Reference to Fig. 72 will show that the 
economic limits of operation are between 3300 kw. and 6300 
kw.; beyond these limits the water rate rises so rapidly as to 
make operation undesirable under this condition, except for a 
short period during peak loads. 

"10. The choice was thus narrowed down to either the high- 
pressure steam turbine or the low-pressure steam turbine. There 
was sufficient space in the present building to accommodate three 
7500-kw. units of the high-pressure type, or a low-pressure unit 
of the same size on each of the nine engines, so that the 
questions of real estate and building were eliminated from the 
problem. 



$m 



THERMODYNAMICS 



"11. The 'first cost of a low-pressure turbine unit is slightly/ 
lower than that of a high-pressure unit, due to the omission of the 
high pressure stages and the hydraulic governing apparatus, 
but the cost.x>f ^he condensing^apparatiis would be. the. same in 
both cases. .The foundations and the.steam piping in both cases 



23 30 



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would not differ greatly.. The economic^ results, so far as the fisst 
cost is concerned, would then be approximately the same, if 
we consider the general case only; but in this particular instance 
the installation of high-pressure turbines would have meant 
a much greater investment for foundations, flooring, switchboard 



STEAM TURBINES 331 

apparatus, steam piping and water tunnels, amounting to an 
addition of not less than 25 per cent to the first cost. 

" 12. The general case of displacing reciprocating engines 
and installing steam-turbine units in their place was also con- 
sidered. The best type of high-pressure turbine plant has a 
thermal efficiency approximately 10 per cent better than the best 
reciprocating-engine plant, but the items of labor for operation 
and for maintenance, together with the saving of about 85 per cent 
of the water for boiler-feed purposes and the 10 per cent of coal, 
reduce the relative operating and maintenance charges for the 
steam-turbine plant to 80 per cent, as compared to 100 per cent 
for the reciprocating-engine plant. 

" 13. Assuming that the reciprocating engine plant is a first- 
class one and has been well maintained, about 20 per cent of its 
original cost (for engines, generators and condensers) may be real- 
ized on the old plant and so credited to the cost of the high-pres- 
sure turbine plant. But on the other hand, if the high-pressure 
turbine installation is to receive credit for the second-hand value 
of the engines, it must also have a debit charge for 100 per cent 
of the original reciprocating-engine plant which it displaced. 
The relative investments, therefore, upon this basis would be 
approximately equal for the high-pressure or the low-pressure 
turbine; but 80 per cent of the cost of the original engine plant 
would have to be charged against the high-pressure turbine plant, 
as against an actual increase in value (to the owner) of the engine 
by reason of its improved thermal efficiency, due to the addition 
of the low-pressure turbine. 

" 14. The preliminary calculations, based upon the manu- 
facturers' guarantees for the low-pressure and high-pressure 
turbines, showed that the combined engine-turbine unit would 
give at least 8 per cent better efficiency than the high-pressure 
turbine unit, so that it was finally decided to place an order 
for one 7500-kw. (maximum rating) unit, as by this means we 
would not only get an increase of 100 per cent in capacity, but 



332 THEKMODYNAMIOS 

at the same time give the engines a new lease of life by bring- 
ing them up to a thermal efficiency higher than that attained by 
any other type of steam plant. 

" 15. The turbine installed is of the vertical three-stage 
impulse type having six fixed nozzles and six which can be operated 
by hand, so as to control the back pressure on the engine, or the 
division of load between engine and turbine. An emergency 
overspeed governor, which trips a 40-in. butterfly valve on the 
steam pipe connecting the separator and the turbine and at the 
same time the 8-in. vacuum breaker on the condenser, is the only 
form of governor used. The footstep bearing, carrying the weight 
of the turbine and generator rotors, is of the usual design supplied 
with oil under a pressure of 600 lb. per sq. in. with the usual 
double system of supply and accumulator to regulate the pressure 
and speed of the oil pumps. 

" 16. The condenser contains approximately 25,000 sq.ft. 
of cooling surface arranged in the double two-pass system of water 
circulation with a 30-in. centrifugal circulating pump having a 
maximum capacity of 30,000 gal. per hr. The dry vacuum 
pump is of the single-stage type, 12-in. and 29-in. by 24-in., fitted 
with Corliss valves on the air cylinder. The whole condensing 
plant is capable of maintaining a vacuum within 1.1 in. of the 
barometer when condensing 150,000 lb. of steam per hr. when 
supplied with circulating water at 70 deg. fahr. 

" 17. The electric generator is of the three-phase induction 
type, star-wound for 11,000 volts, 25 cycles and a speed of 750 
r.p.m. The rotor is of the squirrel-cage type with bar winding 
connecting into common bus-bar straps at each end. This type 
of generator was chosen as being specially suited to the conditions 
obtaining in the plant. 

"18. With nine units operating in multiple, each one capable 
of giving out 15,000 kw. for a short time, operating in multiple 
with another plant of the same size, it is evident that it is quite 
possible to concentrate 270,000 kw. on a short circuit. If we 



STEAM TURBINES 333 

proceed to add to this, synchronous turbine units of 7500-kw. 
capacity, which, owing to their inherently better regulation and 
enormous stored energy, are capable of giving out at least six 
times their maximum rated capacity, the situation might soon 
become dangerous to operate, as it would be impossible to design 
switching apparatus which could successfully handle this amount 
of energy. The induction generator, on the other hand, is entirely 
dependent upon the synchronous apparatus for its excitation, 
and in case of a short circuit on the bus-bars would automatically 
lose its excitation by the fall in potential on the synchronous 
apparatus. 

"19. The absence of fields leads to the simplest possible 
switching apparatus, as the induction generator leads are tied 
in solidly through knife switches, whichare never opened, to the 
main generator leads. The switchboard operator has no control 
whatever over the induction generator, and only knows it is 
present by the increased output on the engine generator instru- 
ments. 

" 20. The method of starting is simplicity itself — the exciting 
current is put on the engine generator before starting the engine, 
and then the engine is started, brought up to speed and synchron- 
ized in exactly the same way as before. While starting in this 
way, the induction generator acts as a motor until sufficient 
steam passes through the engine to carry the turbine above 
synchronism, when it immediately becomes a generator and picks 
up the load. Three of these 7500-kw. low-pressure turbine units 
have been installed and tests run on Nos. 1 and 2. No. 3, having 
been just started, has not yet been tested. 

"21. Instead of inserting in this paper the enormous accumu- 
lation of data incident to these tests, we have divided the paper 
into two parts in the hope that it would thus be more accessible 
for reference, the first part giving the reasons for adopting this 
particular type of apparatus, with a brief description of the plant 
and a summary of the results obtained, and the second part con- 



334 THERMODYNAMICS 

taining all the principal data acquired during the tests, with 
sufficient explanation to make their meaning clear without refer- 
ence to the text." 

" 24. The net results obtained by the installation of low- 
pressure turbine units may be summarized as follows : 

"a. An increase of 100 per cent in maximum capacity of plant. 
" b. An increase of 146 per cent in economic capacity of plant. 
" c. A saving of approximately 85 per cent of the condensed 

steam for return to the boilers. 
" d. An average improvement in economy of 13 per cent over 

the best high-pressure turbine results. 
" e. An average improvement in economy of 25 per cent 

(between the limits of 7000 kw. and 15,000 kw.) over the 

results obtained by the engine units alone. 
"/. An average unit thermal efficiency between the limits of 

6500 kw. and 15,500 kw. of 20.6 per cent." 

205. Summary. The two preceding quotations are self-explan- 
atory; hence no comment is necessary. But, before concluding, 
it must be remarked that the internal combustion engine and the 
steam turbine are still in the experimental stage; and that it is 
impossible to predict what the final adjustment will be. It is 
true that the internal combustion engine has a higher thermal 
efficiency than has any other heat motor. But, due to complexity 
of construction, the internal combustion engine has a higher first 
cost; and furthermore, its regulation is inherently inferior to a 
reciprocating engine or turbine. Due to this, in spite of the fact 
that the reciprocating engine has a lower thermal efficiency, 
it still holds its place, on account of its simplicity and high over-load 
capacity; the latter being especially important in most power 
plants where it is necessary to take care of large " peak 
loads." 

It must be further remarked, that the installation of every 
power plant is finally affected by the economy of transmission. 



STEAM TUEBINES 335 

Whether power can be developed economically at any locality 
depends upon whether the cost of power for the particular locality 
is greater or less if developed at this particular point, or developed 
at some other point and transmitted to the point under considera- 
tion. This, of course, depends largely upon the economy of 
transmission. 

At the present time electrical engineers are giving consider- 
able attention to the subject of high-tension transmission. And 
if it develop that methods can be devised by means of which 
corona losses can be eliminated, or partially avoided, for potential 
differences far in excess of those employed at present, the subject 
of power plant economics will require revision. For, if corona 
losses can be avoided, the cost of power for any particular locality 
will be materially changed. And hence, the cost for the produc- 
tion of power will, likewise, be changed. 

To illustrate concretely: Assume that it becomes possible to 
transmit with a potential difference of 300 kilo-volts instead 
of 125 or 150 kilo-volts. Under these conditions the economy 
of transmission is considerably increased; and the distances to 
which coal can be transported, to compete with the increased 
efficiency of transmission, is considerably reduced. This, however, 
is not the only governing factor. Ground rent also influences 
the choice. That is, when the saving in transmission and the 
saving in ground rent, by locating the plant at the coal fields, 
is balanced against the hauling of the coal, and the ground rent 
for a large city, it may develop that it is more economical 
to locate the power plant where the coal is mined. A similar 
argument, of course, applies to water-power plants. That is, 
the initial cost of a water-power plant is high, and therefore the 
distance, for a given potential difference, over which power can 
be profitably transmitted is limited; and, of course, the lower the 
cost of transmission, the greater the area over which profitable 
transmission may take place. Hence, as the potential difference, 
which may be employed in transmission, is increased, the smaller, 



336 THERMODYNAMICS 

relatively, due to high ground rent, becomes the economy of a 
localized plant. Therefore, if it develop, that potential differences, 
far in excess of those employed at the present time, may be used, 
power plants in large cities, where ground rent is high, will dis- 
appear; and the future power plant will be located at the point 
where the raw material, for the development of power, is found. 



INDEX 



PAGE 

Absolute scale 43 

Absolute zero 43 

Adiabatic changes, change of dryness with 139 

change of temperature with 59 

Adiabatic equation for gases 50 

Air, and adiabatic compression in compressor 235 

and air compressors 230 

composite diagram of compression, transmission, and expansion .... 260 

compression and expansion 231 

isothermal 233 

constants of 230 

loss of head in transmission pipes 254 

Air compressor 230 

throttling and other imperfections 249 

Air motor, adiabatic expansion in 249 

Air refrigerating machines 266 

ideal coefficient of performance of 267 

T-§ diagram of 270 

Air transmission system, theoretical efficiency of 262 

Ammonia, coefficient of absorption 280 

heat of dilution 281 

Ammonia refrigerating machines, absorption 277 

compression 271 

Analysis of power plant losses 310 

Anode 30 

Boyle's law 41 

departure from 43 

Brake power 171 

Brayton cycle 195 

British thermal unit 9 

Calorie - 8 

common 9 

gram 9 

mean 8 

zero 8 

337 



338 INDEX 

PAGE 

Calorimeter 11 

condensing 154 

cooling constant of 14 

Joly's differential steam 70 

Joly's steam 69 

separating 156 

thermal capacity of 13 

throttling 151 

Calorimetry 11 

Carnot's cycle 97 

a reversible process 100 

coefficient of conversion 99, 109 

steam operating on 116 

with a perfect gas 104 

Cathode 30 

Centigrade scale 3 

Characteristic equation , 43 

Chemical reactions of fuels 198 

Clearance 156 

equivalent length of 157 

of air compressor 247 

Commercial efficiency of steam engine 176 

Compound engines, cross 189 

tandem 183, 186, 188 

Compressed air 230 

Compressed air system, composite diagram of 260 

theoretical efficiency of 262 

Compression, adiabatic 235 

isothermal 233 

multi-stage 239 

theoretical efficiency of 237 

Condenser 121 

Condensing calorimeter 154 

Conduction 83 

flow of heat along a bar 86 

Conductivity, coefficient of 85 

determination of coefficient of 86 

in non-isotropic substances 89 

non-homogeneous solids 89 

of gases 90 

of liquids 90 

table of coefficients 90 

Convection 83 

Cooling, constant of calorimeter 14 

Newton's law of 11 

Stefan's law of 82 



INDEX 339 

PAGE 

Critical temperatures, table of 63 

Cross-compound engine 189 

Curtis turbine 307 

Curves, theoretical and actual 163 

Cushion steam 157 

Cycle 76 

Brayton 195 

Carnot's 97 

with a perfect gas as a working substance 104 

Diesel 196 

four phase 193 

of perfect steam engine and boiler represented by means of T-§ 

diagram 137 

Rankine's 177 

reversible, as a standard 103 

two phase 194 

Cylinder feed 157 

De Laval nozzle 298 

Density and temperature of saturated steam 118 

Dew point 66 

Diesel cycle 196 

Disgregation, heat of 62 

Dissociation 28 

Ebullition 61 

Economics of power plants 310 

and transmission 334 

Efficiency and precompression 218 

commercial 176 

mechanical 175 

of power plants and transmission 334 

thermal 175 

Elastic medium, propagation of wave motion 74 

speed of 77 

Elasticities, determination of ratio from speed of propagation of wave 

motion 77 

isothermal and adiabatic and ratio of two thermal capacities . 72 

Electro-chemical equivalent 31 

Electrolysis 29 

counter e.m.f . of 31 

Faraday's statement 30 

Emissivity 81 

Endothermic 21 

Energy, principle of 33 



340 INDEX 

PAGE 

Engine and boiler, perfect, with steam 121 

studied with aid of the T — <J> 

diagram 137 

clearance 156 

internal combustion 192 

actual indicator diagram of 215 

actual p-v and T — c|> diagrams of 222 

double-acting cylinders 227 

multi-cylinder 226 

standard diagram 213 

thermal efficiency of 214 

T-§ diagram of 219 

reversible 101 

and refrigeration 110 

as a standard 103 

ideal coefficient of conversion 99, 109 

simple thermodynamic 95 

steam, condensing 150 

double-acting 150 

Engines, compound 180 

cross 189 

double expansion 181 

tandem, with large receiver 183 

tandem, with small receiver 188 

tandem, without receiver 186 

triple expansion 190 

heat 94 

reciprocating and turbines 308 

reciprocating, combined with low-pressure turbines 310, 328 

Entropy 128 

and temperature diagrams 134 

change of 129 

during reversible and irreversible processes 134 

universal increment of 132 

Evaporation 60 

Exchanges, Prevost's theory of 80 

Exothermic 21 

Expansion, adiabatic, in motor 249 

change of dryness during 167 

double, in steam engine 181 

external heat of 97 

heat of 62 

■ internal heat of .... r 97 

isothermal 233 

triple, in steam engine 190 

without doing external work 46 



INDEX 341 

PAGE 

Expansion, linear 34 

coefficient of .• 34 

factor of 34 

table of coefficients of 36 

voluminal 35 

determination of coefficient of, for liquids 36 

direct measurement of 39 

Fahrenheit scale 3 

Four-phase cycle 193 

Frequency 76 

Friction brake 171 

Fuel, calorific value of 203 

determination of 204 

Fuels, and chemical reactions 198 

and fuel tests 198 

liquid 207 

table of calorific values of 208 

Fusion, heat of 22 

Gas, characteristic equation of 43 

city 203 

perfect or ideal .' . . 44 

thermometer 44 

Gases, adiabatic equation 50 

change of temperature with adiabatic changes 59 

conductivity of 90 

elasticities of 72 

expansion of, without doing external work 46 

Gay-Lussac's law 41 

general equations of 52 

isothermal equation 41 

table of constants 208 

thermal capacities of 47 

at constant pressure 48, 68 

at constant volume 48, 69 

determination of 68, 69, 70, 74 

Gasoline 200 

Gram calorie 9 

Head, loss of, in transmission pipes 254 

Heat, as a measurable quantity 7 

content 125 

effects of 33 

engines or motors 94 

flow along a bar 86 



342 INDEX 

PAGE 

Heat, mechanical equivalent of 17 

from constants of air 48 

of dilution 281 

of disgregation 62 

of fusion 22 

of vaporization 23 

for water 65 

production of 33 

total, of steam 65 

unit quantity 8 

Heating machine 287 

Humidity, absolute 67 

relative 66 

Hydraulic-radius 255 

Hygrometry 66 

Impact on curved surfaces 289 

Indicated power 173 

Indicator 159 

Indicator diagram 161 

and valve adjustment 161 

comparison of theoretical and actual curves 163 

ideal of internal combustion engine 210 

standard of internal combustion engine 213 

Internal combustion engine 192 

actual indicator diagram of 215 

actual p-v and T — <£> diagrams 222 

and double-acting cylinders 227 

efficiency and precompression 218 

multi-cylinder 226 

standard diagram of 213 

thermal efficiency of .• 214 

Irreversible processes 27 

Isothermal equation 41 

Isotropic 35 

Kerosene 201 

Mean effective pressure 174 

Mechanical efficiency of steam engine 175 

Mechanical equivalent of heat 17 

from constants of air 48 

Mixtures, method of 10 

Motor, adiabatic expansion of air in 249 

reheating air before expanding in 251 

Motors, heat 94 



INDEX 343 

PAGE 

Multi-cylinder internal combustion engines 226 

Multi-stage air compressors 239 

Non-isotropic 36 

Parsons turbine 306 

Pelton cup 293 

Power, brake 171, 214 

indicated 173, 214 

Power plant economics 310 

analysis of losses 310 

efficiency and transmission 334 

Precompression and efficiency 218 

Pressure and temperature of saturated steam 119 

Pressure, mean effective 174 

Priming 126 

Principle of energy 33 

Radiation 80 

Dulong and Petit's formula 82 

Prevost's theory of exchanges 80 

Stefan's formula 82 

Rankine's cycle 177 

Reaumur scale 3 

Reciprocating engines and turbines 308 

engines exhausting to low-pressure turbines 310 

Refrigerating machine 110 

absorption 277 

air 266 

ideal coefficient of performance of 267 

T-cJ> diagram of 270 

commercial 265 

commercial efficiency of 269 

comparison of air and ammonia 284 

compression, using volatile liquids 271 

T-4> diagram of 276 

Kelvin heating 287 

Refrigeration 265 

and reversible engine 110 

Refrigerator 94, 266 

Resisted adiabatic expansion of steam 126 

Reversible process, ideally 27 

Reversible processes 25, 104 

Separating calorimeter ' 156 

Specific heat 9 



344 INDEX 

PAGE 

Steam, behavior throughout the cycle 166 

change of dryness during adiabatic expansion 126, 139 

by means of T — <f> 

diagram 139 

cushion 157 

cylinder feed 157 

double expansion engine 181 

dryness during expansion in cylinder 167 

engines, reciprocating and turbines 308 

engines, reciprocating with low-pressure turbines 328 

exchange of heat with cylinder walls 168 

jackets 170 

operating on Carnot's cycle 116 

relation of temperature and entropy 139, 141 

resisted adiabatic expansion 126 

saturated, density and temperature of 118 

relation or pressure and temperature 119 

total heat of 65 

triple expansion engine 190 

turbine 289 

unresisted adiabatic expansion 125 

wire drawing V 158 

with perfect engine and boiler 121 

work and incomplete expansion 145 

work and superheating 148 

work without expansion 144 

zero curve 142 

Zeuner's equation for adiabatic changes 151 

Stefan's formula 82 

Sublimation 23 

Superheated vapors 66 

Superheating 24 

Tandem compound engines 183, 186, 188 

Temperature 1 

critical 62 

table of 63 

difference of . : 2 

Temperature and density of saturated steam 118 

Temperature and pressure of saturated steam 119 

Temperature-entropy diagrams 134, 219 

dryness by means of 139 

of air refrigerating machines 270 

Temperatures, theoretical in cylinder of internal combustion engine 213 

thermodynamic scale of 113 

Thermal capacities, determination of by method of cooling 16 



INDEX 345 

PAGE 

Thermal capacities, determination of by method of mixtures 15 

Thermal capacities of gases 47 

determination of, at constant pressure 68 

at constant volume 69 

ratio of, by method of Clement and De- 

sormes 70 

ratio of, from speed of propagation of wave 

motion 77 

Thermal capacity 8 

by method of mixtures 10 

per unit mass and specific heat 9 

Thermal efficiency of internal combustion engine 214 

of steam engine 175 

Thermal equilibrium 2 

Thermo couple 6, 32 

Thermodynamic drop 158 

scale of temperatures 113 

Thermodynamics, first principle of 93 

second principle of . 93 

Thermometer 2 

alcohol 6 

gas 44 

mercurial 5 

resistance 6 

Thermometric scales 3 

Throttling and other imperfections of air compressor 249 

calorimeter 151 

Transmission of air and loss of head in pipes 254 

of power and economy of plants 334 

Turbine blades, impact on 289 

Turbines, and reciprocating engines 308 

comparison of Parsons and Curtis 307 

Curtis .307 

low pressure on exhaust of reciprocating engine 310, 328 

Parsons 306 

steam 289 

tests 310 

two principal types of 305 

Two phase cycle 194 

Unresisted adiabatic expansion of steam 125 

Valve adjustment and indicator diagram 161 

Vaporization 60 

heat of 23 

heat of, for water 65 



346 INDEX 

PAGE 

Vapor pressures, addition of 61 

Vapors 60 

saturated 60 

superheated 24, 66 

Water equivalent 10 

Water, relation of temperature and entropy 137, 141 

Wave length 76 

Wave motion, propagation of 74 

speed of propagation 77 

Wire drawing 158 

Work, gain of, due to superheating steam 148 

loss of, due to incomplete expansion of steam 145 

loss of, due to using steam non-expansively 144 



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